People have answered how the actual process works, so let me answer your other question:

> how would someone be able to tell that the money wasn’t legit?

They can't, directly; cash isn't stamped with a "DRUG MONEY" mark or anything, after all. What they can tell is that you don't seem to have a reasonable explanation for how you legally acquired it.

Say I sell drugs or whatever and make millions of dollars in cash. I can use it on groceries and restaurants and the like, and I can use it to pay other people for illegal things (like more drugs to sell), but at some point I will want to use it to buy a house or a car or whatever large purchase. Buying expensive things with physical money is super weird and looks suspicious, and people will probably report me to the government if I try.

So I want to pay electronically, right? Well, how do I get my big stacks of drug money into the system? If I just take a million dollars to the bank, they'll report it to the government (they have to by law), and the government will have a lot of questions about how I got it and I won't have a good explanation. I could break it up into smaller amounts and deposit it over time in an attempt to evade detection (a process known as structuring), but banks know about structuring and I'd have to deposit it over an enormous amount of time in order to not looks suspicious.

So instead, I start up or purchase a business that operates in cash - something like a laundromat, for example. Now I have a perfectly plausible reason for depositing thousands of dollars in cash each week: that's how my "customers" pay me. This gives me a way of getting it into the banking system, and once it's in the system I can buy my house or car or boat or whatever just like any normal person. Sure, I lose some of the money in the form of taxes and whatever I have to spend actually running your business (having a front does require some actual cash outflow), but that's a small price to pay compared to having an enormous amount of money sitting in a warehouse that I can't use on anything.

Now to be fair, there's a second alternative. If I simply don't spend or deposit very much of the money, or if I use it as cash over a very, very long amount of time, nobody will have any way of figuring out I got it illegally; the problem is that what's the point of doing risky, illegal things in order to get money you can't use?