You can find the full details here, but in short, you can set a callback URL for your app at http://www.box.com/developers/services that Box will send the authentication token to. You can set up your web service to receive the token there.
The way Box's authentication flow is designed is to prevent the need for users to provide 3rd parties (you in this case) with their credentials. The fact that there is no mechanism to authenticate a user by passing their credentials through a web service call is deliberate, and there is no straightforward work around I'm aware of (especially no such workaround that would be consistent with the terms of use of their API).
http://developers.box.com/get-started/#authenticating
There is a new authentication mechanism they have in the pipeline which will allow you to authenticate to a sandboxed folder for your application using a much simpler process. This is currently in private beta. This might fit the needs of what you are looking for once it is available.
