I'm sure op is not asking about mining, he is asked how anyone can get BTC stayed as anonymous, It is not possible for everyone who wants to remain anonymous to start mining Bitcoin.
The simplest way is buy Bitcoin and use a mixer to mix your coin. If you use Chipmixer then they will give you an equivalent amount of chips before you make your deposit, so you will able to stay anonymous cause no one will find the relation between your chips and deposits.