I think it helps to start with the idea that Trump lives in a world where you can do whatever you want to people as long as you have somebody else come along and “take care” of them. Who knows how many non-disclosure/settlement agreements he’s reached over the years to avoid the embarrassment of being publicly outed for his conduct? (He’s certainly made mention of such contracts)
Once the election got into full gear, it became even more important for Trump to silence these stories. As somebody who had a track record of such things, he just had to ramp up the intensity of his usual cohort of “fixers”. I can imagine him yelling into the phone “find all of them! I don’t want anything getting out!”
So, his group of henchmen (who themselves likely only served their role because of how lucrative it was for them) began to gobble up whatever they could.
And since it was a group of people (some - like Pecker - in the media, and others - like Cohen - having law licenses) who had experience with this sordid stuff, they were easily able to work together to “catch and kill” the stories.
If one went to a lawyer, one of Trump’s “lawyers” would get wind of it and swoop in to “represent” them. And if one went to a media outlet, Trump’s “media” team would swoop in to provide them a “forum”.
At that point, it’s just a matter of paying money in exchange for complicity.
But, now you have a possible political scandal on your hands (in conventional times, can you imagine the fallout of a candidate having paid off a mistress?), so you have to be sure to obscure your actions.
However, to do that, you have to set up fake companies with fake invoices, since you can’t just withdraw hundreds of thousands of dollars in cash without being asked what it was for.
Thus, the effort to hide immoral behavior dovetails into criminal behavior. The lying is criminal, and the lying to influence the election (or, taking advantage of the lying by pretending it is real, and therefore a business write off) elevates the criminality to a felony.
And here we are.