The anonymous aspect is that when you mine Bitcoins you can do so anonymously. It isn't very practical any more as you need to be part of a pool to stand much chance of actually earning many, but in theory you can create them in an untraceable way (with the usual caveats about using Tor properly etc.)
You can also spend them anonymously if it is only on stuff that doesn't require anyone to know your identity or interact with you, so basically donation. Might indirectly benefit you, such as a fund to have someone you don't like assassinated. For the most part though it is hard to actually use Bitcoins anonymously. You also have the problem of Bitcoin addresses being linked, so when mining you need to keep creating new addresses with each new coin you mint, and ideally spend them in exact multiples of one since spending 0.5 BTC twice from the same address links the two transactions.