Learn English – The difference between “wait up here” & “wait here”

OED 1 gives this definition:

6. A smartly-liveried boy acting as groom or footman; formerly often provided with standing-room on a small platform behind the carriage, and a strap to hold on by; less strictly, an outdoor boy-servant. slang. obsolescent.

The citations given here suggest that the term became current about 1817. OED offers no speculation about why the term was adopted. My guess is that it reflects the very colorful liveries which were popular in the 19th century. Livery is the formal uniform which wealthy people used to require their servants to wear. Livery was always markedly different from ordinary dress—you don't want your servants to be mistaken for your peers! As a form of 'conspicuous consumption' it was usually patterned on the formal dress of an earlier period and executed in bright, eye-catching colors, which contrasted particularly in the 19th century with the sober colors preferred by gentlemen. Here is an example of the livery worn by Queen Victoria's coachmen; note the powerful stripe effect:

@Stephie notes a source which reports that “The tiger wore an orange and black striped waistcoat which is what gave them their ‘title’.” It may very well be that the first footboy to be called a tiger wore such a waistcoat, and even that such a waistcoat was at some point the 'default' dress for footboys, just as green-and-black stripes were common for table-servants in the early 20th century. But I have been unable to find a contemporary illustration. It should also be noted that the ancient practice was that liveries were executed in colors specific to the family, so it cannot have been a universal practice.

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...".