I'm reading papers since hours and still don't get one thing: A bitcoin transaction is an information like "Bob sends 2 BTC to Alice". This message is signed with Bobs private key and needs to be added to the blockchain that is distributed to every participant. I also got that you need to calculate a Proof of Work to be able to add something to the chain. Or in other words: The other participants won't accept your block without the proof of work.
How does Bob get this proof of work or how does Bobs message went to a person who has such I proof? I can't believe that every person making a transaction needs to calculate it's own proof. Is it really like that? If not: How does Bobs "message" is transmitted to a person who owns such a proof, how can the person make sure that Bobs message is a valid transaction (and not only garbage) and what the motivation for this person to process Bobs message at all?