Once the IdP knows who the user is, it sends the user's browser back to the SP with a SAML Authentication Assertion message.
This message contains no real information about the user but does assert that the IdP has successfully identified them, how it did so, when it happened, etc. The IdP also copies the URL that the user originally requested into the response from the request.
This message can't be sent as a HTTP Redirect because it is too big, so it is sent by embedding it in a form field and then using JavaScript to automatically submit the form (or getting the user to click a button). The form is submitted to the SPs Assertion Consumer Service (the location of which was also included in the authentication request).