If you are acting as the proxy for the user, then all web requests go through you. A normal proxy would just forward those on to the Amazon (e.g.) server.
In your scenario, the proxy sends a cert to the client saying 'I'm Amazon', and the client believes it. The client talks to you like you're Amazon, and you read all his messages. At the same time, you talk to Amazon pretending to be the user, so you can get all the correct Amazon responses to send back to the user.