The basis for understanding Latin, Greek, etc. roots is to help you make an educated guess at the meaning of new words you encounter.
Often, context will give you sufficient clues to guess the word's meaning, but not always. If you are familiar with roots and you can recognize the root of the new word, you can get a pretty good idea of its meaning even when there is not enough context to clearly define it, or when the context is ambiguous. Roots are especially helpful when used in conjunction with a word's context.
It also helps to know the English roots of words, so that when you come across compound words or words with crazy prefixes and suffixes you can still pretty well guess what someone is talking about.
Not to get overly philosophical, but it is the nature of language that there is no universal, eternal, provable right answer.
If we were debating a scientific question, then, at least in principle, we could perform an experiment and find the right answer. If person A says that the chemical formula for water is H2O and person B says that it is N2O, person A is right and person B is wrong, period.
But in language, if an American says that a certain word is spelled "color" and a Briton says that it is spelled "colour", there's no experiment we can do to prove that one is right and the other is wrong.
Maybe other languages have some single recognized authority who declares right and wrong usage, so if in doubt you can check that book or ask that institution and get the official right answer. But even at that, I'm sure their answers would change over time as, for example, new words are added to the language to describe new ideas or new inventions.
Regardless, there is no single recognized authority in English. There are a number of highly respected authorities. The Oxford English Dictionary is highly respected for definitions of words. The Chicago Manual of Style, the Modern Language Association Handbook, and Strunk and White's Elements of Style, are all widely respected.
A key element in the differences of opinion comes down to how you decide what the rules are. I think all serious language students agree that what is actually used by most speakers of the language is an extremely important element. If 99% of the people use a word with a certain definition, it is pretty meaningless to say that that is not a correct definition of the word. Again, it's not like science or math: If you took a poll and discovered that 99% of the people agree that the Earth is flat, sorry, it's still round. But if you took a poll and discovered that 99% of the people agree that the word "flat" means, I don't know, "made out of wood", then that is what it means, because that is what everyone agrees that it means.
A problem arises when new words are in the process of being invented, or when the accepted definition of a word is changing. Then there can be a period when 50% of the people think a word means X and the other 50% think it means Y. To take a recent example, if you asked people 50 years ago what the word "gay" meant, they would say "happy and carefree". But today it means "homosexual", and if you use the word in the old sense people will at best find it an amusing out-of-date usage, or at worst misunderstand your meaning.
Some language exerts say that common usage is the only criteria for correctness. Others -- and I agree with this second group -- say that logic and consistency are also valid criteria in deciding what is "correct". To take a current example -- not the best example but an easy one to describe -- it is becoming increasingly common in English to use "they" as a gender-neutral singular pronoun. Others say that this is wrong, that "they" is plural and should only be used as a plural because otherwise we lose the distinction between singulars and plurals. If a rule helps avoid ambiguity or assists clarity, these folks may say that it is a good and valid rule that SHOULD be obeyed even if a majority do not obey it.
And let me add that there is a third group that says that something is a rule because somebody at some time wrote it in a book or taught it to them in school, even though it is neither commonly accepted nor particularly logical. A good example of this is the rule, "Never end a sentence with a preposition." Few serious linguists recognize this as a valid rule because, (a) many English-speaking people, including many well-educated and literate people, break this "rule" all the time; and (b) It serves no rational purpose.
Best Answer
Latin grammar
Latin has an inflected grammar, in which words change their form to indicate the role they're playing in a sentence. English has a little bit of inflection; Latin has a lot. For example, in English, these are all the possible forms of a verb: show, shows, showed, shown, showing. Most Latin verbs have about 150 different forms. These indicate how the verb fits into the sentence, which noun it agrees with, and other things.
English has a possessive case for nearly all nouns, indicated by -’s, and distinguishes two other cases only in a few pronouns: I/me, he/him, she/her, they/them, and (sometimes) who/whom. Latin has five cases (or six, or seven, depending on how you count them), and they apply to all nouns. For example, in the Latin sentence Marcus Quintum pulsat, which means “Mark hit Quintus”, the -us ending indicates that Marcus is doing the hitting, and the -um ending indicates that Quintus is the person getting hit. The word order is very flexible, because the cases indicate how the sentence fits together. You could say Quintum pulsat Marcus and it would mean the same thing.
In Latin, all the grammatical distinctions and categories are explicitly indicated by the inflections. The meaning of most sentences results from combining the meaning of each individual word according to the grammatical rules (known as “the principle of compositionality”). It’s easy to see why grammarians would like to use the concepts of Latin grammar to explain the grammar of all languages. Latin has rules, which are mostly clear and obvious.
English grammar
But English grammar works very differently from Latin. English grammar is more of a patchwork of phrases, with a few inflections to help out (like -ed, -s, and -ing). Many phrases have unique rules for how to use them, like “too adjective to verb”. Grammatical distinctions are often indicated by ambiguous auxiliary verbs like will, would, and should, whose meaning varies a great deal depending on the phrase or the context. Here is a list of verbs, some of which can be followed by a gerund, some of which can be followed by an infinitive, and some of which can be followed by either. There is no rhyme or reason to the list.
Unlike Latin grammar, which the Ancient Romans had written about and explained in detail just from noticing when they use each inflection, English grammar is quite hard to figure out and explain. Even today, linguists are still debating over what the grammatical categories are. For example, sometimes there seems to be no way to tell which (Latin-like) grammatical category is denoted by -ing, as in this question. Probably the most complete attempt to formulate English grammar as a set of rules is this book, which is 1,800 pages long and costs US$250.00!
The effect on education
For a couple hundred years, many educators thought you could understand English grammar by learning Latin grammar. It was thought that English grammar was doing in an “abstract” or “invisible” way what Latin grammar does “concretely” or “visibly” with inflections. So, learning Latin grammar seemed like an easier way to learn the grammatical concepts of English. For example, Dorothy Sayers (1893–1957) said “To embark on any complex English construction without the Latin Grammar is like trying to find one's way across country without map or signposts.” The idea that Latin and English grammar do not just represent the same things in superficially different ways, but work in profoundly different ways, so that the “map” you learn from one language misleads you in the other language, did not seem to occur to many people. The assumption that, deep down, English and Latin grammar work the same way, is much less popular now than it used to be, but a few people still believe it. (This assumption might even still influence linguists of the type who try to work on a “universal grammar”.)
Consequently, some of what most Americans are commonly taught about English grammar is wrong or confused, because American schoolteachers’ grammar concepts are largely borrowed from Latin.
For example, English verbs have an -ing form, which doesn’t neatly correspond to anything in Latin. Some people are taught that the word running in “I am running” is a present participle, and others are taught that it’s a gerund. Present participle and gerund are concepts from Latin grammar. However, in the English sentence “I am running”, neither of the Latin grammatical concepts fits well. Yet, schoolteachers teach the Latin grammatical terminology as if it applied to English grammar. Mostly this causes confusion. The schoolteachers themselves probably don’t even know they’re doing this.
Two famous errors, “taught” up until about twenty years ago, were that it’s wrong to end a sentence with a preposition, as in “Jane picked the cat up”, and wrong to “split” an infinitive, as in “to boldly go”. These are natural, grammatical constructions in English. However, under the influence of Latin grammar, many educators thought they were ungrammatical. The problem is that phrases like pick up and to go are two-word units of meaning in English grammar unlike anything in Latin grammar.
In Latin, a preposition always introduces a prepositional phrase, which includes a noun. If you say in in Latin, you always follow that by saying in what. For example, Marcus Quintum in via pulsat, “Mark hit Quintus in the road.” In Latin, it really doesn’t make any sense to end a sentence with a preposition. No one would do that, just as in English, no one would end a sentence with an article.
And Latin has nothing corresponding to the to-infinitive of English. It’s not possible to split an infinitive in Latin even if you wanted to. Many people learning English as a foreign language have been misled by being told that to verb is how you make an infinitive in English, by false analogy with Latin infinitives. Hence “
I go for to buy cigarettes.”To this day, most Americans do not know what phrasal verbs are, even though they use them every day. And many people learning English as a foreign language are never taught about phrasal verbs, because phrasal verbs don’t correspond to anything in Latin grammar. Lack of a name or even a concept for phrasal constructions causes a lot of confusion for people whose native language doesn’t have them.