At a fundamental level it's pretty simple:
Have account A in currency Y. Have account B in currency X.
When account A receives funds, release funds from account B.
This is how companies like wise.com work.
The complicated part is the regulation and reporting needed for cross-border transactions.
To be fair to crypto, it does offer one way of disconnecting the two accounts - rather than having any trust between them, you can simply use a trusted distributed ledger. This in theory creates a marketplace. (Keeping in mind that any trusted system could achieve the same goal, just happens that you can do this with crypto). Of course, it does so by adding cost, complexity, and maybe making your transactions public.
The real 'benefit' is that because crypto is new, and wild west, is that there's no guarantee that accounts A and B are complying with regulation and reporting needed for cross-border transactions (or even banking in general). As such on the surface it can seem like this is a cheaper and easier alternative - but only because it's avoiding the problem, not solving it. Given enough time and countries will likely crack down on this.
And sure, the countries are already cracking down on it, just as they've been cracking down on Hawala before. So far, such attempts to regulate seem to have the most effect on miner operations and speculation, not on meaningful movement of money.