Steam Trading Error – “This trading session has expired.”

This means that - for whatever reason - the built-in browser handling these pages somehow lost the data identifying the current session (which is essentially just some semi-random value to identify the computers allowed to actually see the data and modify it; i.e. those that are allowed to complete the trade).

There are several reasons why this could happen:

• The whole trade session might have been open simply too long. For example, you might have opened the trade and then didn't try to complete it "in time". I don't know how long that time is, but I'd assume at least a few minutes (unlikely more than an hour).
• Some server handling the request(s) dropped that Information. This could be a local proxy or just some server at Valve acting up.
• Your Internet connection dropped for a few seconds while you tried to perform the trade. In this case you might have been assigned a new IP by your ISP, which most likely would invalidate the current session for security purposes.
• If you're using some kind of firewall or filter software modifying the content of web pages (to "protect" you or remove ads), this might be the reason as well (filtering this legitimate data intentionally or by accident). Try to add the Steam client to its exception list.