"Balancing the check book" refers to the process of running your own log of check transactions using something like
https://cdn.wallethub.com/wallethub/images/posts/14483/check... and ensuring that it matches the occasional official balance statement from your bank occasionally.
Don't want to make assumptions about what you know so forgive any overexplanation...
It is about ensuring that you manage your account balance, so as to not write checks that your account balance cannot actually fulfill. Writing a check that you can't actually pay because your checking account is too low is called "Bouncing" a check, and generally costs you money at your bank and the place that you wrote the check, and will often result in those places being less likely to accept a check from you.
A secondary purpose was to ensure that your bank did not pay out a check that you have no record of writing. The check system has almost no protection, so anyone who had ever gotten a check from you could theoretically attempt to create fraudulent and forged checks against your account. You would check your record against the banks to check for this.
You would do this constantly, as in writing down every check you write, and once a month or so, actually doing the math, comparing your record of checks against the bank's record of checks.
I would also consider it a euphemism for "Financial literacy" in plenty of people's heads.
It is without a doubt the simplest possible mathematical task you can have as an adult. It is literally addition and subtraction, and logging every check. Every single American who attends public school until 4th grade has been taught the required knowledge for this task: Addition, subtraction, and the concept of negative numbers.
People say "Schools should teach balancing a check book" as some sort of cry that they think schools should focus on "practical" skills rather than, say, persuasive essay writing or reading literature or learning art.
Those people are demonstrating that they are too stupid to even understand what they have been taught. These same people will sit in math class and argue about how they "will never use this", in reference to things like calculating interest. They will often also say "Schools don't teach critical thinking", openly and proudly ignorant of the hilarity of someone who can't follow very basic instructions that anyone can find in five seconds and often come with the checkbook complaining about not being taught how to think.
Similarly, Americans will complain that school didn't teach them how to "do their taxes", which is hilarious, because for 95% of Americans, your federal taxes are a couple of pages and about 14 lines of numbers you have to fill in, most of them are numbers you copy from another piece of paper, and the rest have literal worksheets to follow.
Most people joke about how VCRs used to constantly have the "12:00" blinking clock because nobody would set them because it's "too hard". As a literal child who fixed this exact issue, it was a single page of instructions, and they were trivial. But for the majority of Americans who like to speak up, it seems like they are totally incapable of following even the most basic of instruction.
Frankly, stupidity is not a moral failing. Nobody chose to be born stupid. But ignorance, especially fixable ignorance, IS. America has a serious problem of people being openly ignorant and thinking their ignorance should be anything other than an opportunity to improve. I think the biggest problem in the USA is how much respect and attention people who demonstrably refuse to learn and are ignorant of pretty much everything get. That's why, for example, you have people insisting that rail transport is impossible in the US because it's too big, despite the fact that we literally built a cross country rail network using public funds when the country was dramatically less populated than now, or how government inherently mismanages healthcare despite about 100 examples of government mismanagement producing better outcomes than we get.