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.
"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...".