There are multiple ways to see the changes of a game, each one of them shows different information:

Changelog for recent updates

The Your Games & Software section of Recently Updated shows simple changelogs written by the developers.

Changelog for old updates

If you want to search for old updates, you can try the News section of a game, with the filter of Official Announcements enabled.

Technical information (size, files changed, hashes, etc.)

If you want the technical information of the update on the other hand, you can try SteamDB. First, go to https://steamdb.info and search for the application or game that you are looking for. After that, you should see a window like this one:

On the top right area, click Patchnotes. You should see something like this:

Over there you can select the day of the update to get the technical changes of it, like the manifest, build id and the files that got changed: