The password is indeed used to encrypt your files (although indirectly, which was the case in AxCrypt 1.x also). Technically it’s used to encrypt a “session key”, a unique, per file, completely random key.
It is never stored on the server, and the connection to the server is always encrypted, so while the password in fact does travel over the Internet it does so safely.
As for sending files to other people, you’re supposed to use the much more convenient “key sharing” feature of AxCrypt 2. No need to share the password then. Just add the recipients e-mail address to the list of users who should be able to decrypt the file.