>how could they find out if I got that bitcoin legal or not?
Depends on which end of the money flow you're being investigated from.
Example scenario #1: LEAs compromise a darknet market and trace payments for illegal goods/services to one of your public keys. They see subsequent transactions involving those public keys and a company that exchanges BTC->USD (or GBP or EUR or whatever). They get a warrant for information about the exchange from the exchange company. They then have information about your bank and deposits. They get a warrant for information from the bank. They now have a chain of transactions tied to you from sale of illegal goods/services through to your deposit in your name at the bank. If you tumble your bitcoins, theoretically, it's more difficult to make the connection from the public key that received BTC for illegal goods/services and the eventual conversion from BTC to fiat currency in your name.
Example scenario #2: You deposit at your bank an amount of money or in a pattern that raises red flags. LEA gets warrants from the bank and the depositing company (the BTC -> fiat exchange) and gets information about the BTC address involved in the transaction. They then try to tie the BTC address to receipt of BTC for the sale of illegal goods/services. If you're using a DNM that has its payment system set up properly, this should be really difficult. That's a form of tumbling in itself.
LEA ability to associate illegally acquired Bitcoins with you depends on 1) associating an illegal transaction with a given Bitcoin address, and 2) associating that Bitcoin address with you. Tumbling your Bitcoins attempts to make that difficult by obscuring the relationship between the Bitcoin address involved in an illegal transaction that the one that eventually is used to transfer Bitcoins to fiat currency in your name.
