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We can look in our side view mirror, which will give us a good view of what’s coming up along the left side of car, and once that checks out and we see that we have room to pull out, the next rule of thumb is something that is absolutely mandatory to doing this properly and safely is to turn your head.
Why did the speaker say "checks" rather than "check"? What does that checks out mean? Specifically, what is the word that referring to?
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 speaker is being very informal, and the situation is clear enough that he can leave out a certain amount, so he has done so. In this case, "something checks out" means that a statement or proposition has been investigated and found to be true.
We can look in our side view mirror, which will give us a good view of what’s coming up along the left side of car,
What is unstated, but is quite obvious, is that we are not only checking the traffic to our left, but we specifically require that there be nobody coming. This can be considered as a tentative statement that nobody is coming on our left, but one which needs to be verified before going any farther.
and once that checks out
and we have confirmed that nobody is on our left
and we see that we have room to pull out,
we can then pull out.
The phrase could be replaced by, "and once we have checked that there is no one approaching on our left", but "once that checks out" is shorter and easier.
