I am by no means an expert, nor a cryptographer, but I would be interested in knowning a little bit more than what happens at the end of the following chain of events.
What happens at the end of the following chain of events of course is related to what happens first, so please correct me if there are mistakes or incomplete parts in what follows!
Suppose we are in the situation where Alice (A) wants to transfer one Bitcoin to Bob (B).
A's private key has a corresponding public key. The private key is used for the signature. This, together with hashed information relative to the previous chain block, the timestamp and other elements, constitutes the transaction.
The transaction is broadcasted to the network, where it waits until it's validated by the miners (together with other transactions).
In extreme synthesis, what miners do is to try to find a nonce that, added to the transaction data they receive and after being double-hashed, is satisfying a certain difficulty condition (set so that on average a new block is added, say, each 10 minutes).
Once a miner finds a working nonce, she sends it to the network so that it can be (easily) checked. In practice, since it takes time and effort to find that working nonce, this constitutes the proof of work.
What I find hard to understand is: how finding a random number such as a nonce can be proof that the transaction was effectively of one Bitcoin and from A to B?
And how can the nonce be the solution of the problem if no one but A knows about her private key?
In other words, I understand that it's extremely hard to find a working nonce, so that finding a compatible one is effectively proof that a lot of time and energies were spent to find it. I don't understand how the solution to the mathematical problem of finding the nonce relates to the fact that the initial A to B transaction is valid (as opposed to be a wrong or "dishonest" one).