The idiom "to check out" has a lot of meanings, but here it means sense 10: to be confirmed. You are performing a check using the side mirror; you are checking the mirror for traffic. The use of the idiom here is fairly informal and imprecise, but not wrong.
The use of "checks" rather than "check" is simply subject-verb agreement with the indicative pronoun "that," which is the subject of the clause.
I check.
You check.
He/she/it checks.
"That" refers to your check in the mirror; i.e., the absence of cars in the driver's blind spot. Basically, it refers to the sentence's entire first independent clause, which is everything before the first "and."
Having answered your specific question:
Wow. The quoted sentence is a doozy. First of all, it is overly complicated and should be broken up into separate thoughts. Secondly, the second independent clause is ungrammatical:
and [...] the next rule of thumb is something that is absolutely mandatory to doing this properly and safely is to turn your head.
After hiding various modifiers, the stripped-down version looks like this:
and the rule is something is to turn.
The author appears to have completely forgotten that they already wrote the first predicate, "is something ...", when they wrote the second predicate, "is to turn...".
The meaning of "spring in one's step", as Manny used it, is this:
Wiktionary
spring in one's step (idiomatic)
enthusiasm, energy or a positive outlook or cheerful attitude.
After her promotion, she carried out her new position with a spring in her step and a contagious smile.
The meaning of "spring" is "jump" or "leap". Someone who has a spring in his step is the opposite of someone who is depressed.
However, Jay has interpreted "spring in my step" to mean that Manny's peers have told him he has a spring in his step because they think he is effeminate, or even homosexual. Manny didn't mean it that way at all, and "spring in my step" is his own phrase, used to mean that he is light-hearted because of his infatuation with a girl. The joke is Jay's misinterpretation, contrasted with Manny's unconcern.
Compare "light in the loafers":
Grammarphobia
Cassell’s Dictionary of Slang (2d ed.) has only a brief entry, describing the expression as ‘50s American slang and adding that “the image is the stereotyped effeminate male, tripping along.”
The Random House Historical Dictionary of American Slang, which defines it as effeminate or homosexual, lists a series of references for the expression dating from 1967 to 1996.
Best Answer
Rather than grouping "be out" together, it is probably easier to understand if you read it like "[You will be] [out there] [driving safely again]".
By "out there", he means, "on the road". "Out there" could refer to any location, though. You could say, "Get out there and have an adventure!" if you were telling someone to get out of the house and go do something.
As a side note, whoever wrote this driving guide was speaking very informally. There are a lot of grammar mistakes. It's good to practice reading this kind of writing, but don't worry too much if something doesn't make sense.
As another side note, if you say that a person "is out" about something, it means they had a secret, but now everyone knows about it. Therefore, to "be out" means that people know about your secret. You hear this phrase most often when someone who is a homosexual tells people about it for the first time. When I read the title of your post, I thought it would be that kind of question, haha. :)