The verb to enter, apart from other uses that it might have, is a transitive verb that takes a direct object and does not require the use of a preposition like into. You always enter something. That something is usually some kind of place such as a room, house, building etc. For example:
Although it was dark inside, I entered the room.
However, do not confuse the transitive verb to enter with the phrasal verb to enter into something which itself can have a number of slightly different meanings. Though, to tell you the truth, all those meanings are just minor variations on the main one: to be an important aspect or factor of a particular situation. For instance:
It was only after my miraculous survival of the car crash that nearly got me killed that God finally entered into my life.
For more information on how to use this phrasal verb, click on the blue link that I provided as part of my answer.
It is quite common in English to say 'x jumped on y' when you really mean 'x jumped onto y'. So, if I say, 'The cat jumped on the chair', it is highly unlikely that I mean that the cat jumped up on down on the chair. It is almost certain that I simply mean,'The cat jumped onto the chair'.
Similarly, it is highly unlikely that someone will stand on a horse and jump up and down on it, unless you are at a circus performance. 'Jump on a horse', most likely means, 'Mount (or climb onto the back of) a horse'.
There are exceptions. If I said, 'The children are jumping on the bed', then it is very likely that the children are jumping up and down on a bed.
So, coming to your question, why do we say 'Jump on the bandwagon', when we really mean, 'Jump onto the bandwagon'? There are probably two reasons, apart from what I said in my very first sentence.
(1) This is a figurative expression, which really means that you waited until an idea, activity or viewpoint became popular before you showed an interest in it. So you are not really jumping onto anything, let alone a bandwagon. As a consequence, it doesn't matter whether we use 'on' or 'onto'.
(2) This is just a common English idiom, and like most idioms, people use it without giving much thought to its actual underlying origins, i.e., people physically jumping onto a circus bandwagon. Because they don't think about the origins of the idiom, it would be easy for the 'to' at the end of 'onto' to be dropped over time.
Best Answer
This doesn't mean the same as "he went and walked on the roof." This means that he walked from another place to the roof. You would be most likely to walk onto the roof from a scaffold or cherry picker or similar. If he was walking while on the roof, you might say:
"Onto" expresses movement from one place to somewhere else, where you end up on what you were moving towards.
I disagree with Ast Pace's answer on this sentence. While it can mean that you moved to be on top of the tree, I would expect most people to interpret it to mean that you are now holding on to the tree, suspended above the ground but not in the branches. This picture gives a good example.
If you wanted to express that he climbed among the branches of the tree, you could say:
You are correct on this sentence. It means that she entered your room by walking. However, you wouldn't say "entered in" your room – the room is what is being entered, not something inside your room.