Say I have this pull request and I want to download it as if it was its own separate project. How do I go about doing that? I don't see any button for that functionality.
You can download a snapshot of the tree at that commit over here. This is an exported tarball so you won't have any history. Is that what you're looking for? You can get to this by first looking at the commits he wants you to pull and then picking the latest one in the list. Navigating to this URL will give you the diff (i.e. it's examining the commit object rather than actual tree). You can now simply change the commit in the above url to tree or click on the "Browse code" button. Once you do that, there's a "Download ZIP" button on the right which allows you to download the tree.
If you want complete history, then you need to fetch mlwelles changes. You can do this by going to the mlwelles:master repository over here and adding that as a remote to your own local clone using git remote add mlwelles git@github.com:mlwelles/AFOAuth2Client.git. Then you can fetch the changes he's asking you to merge using git fetch remote master. The changes will be available in FETCH_HEAD. You can either view them using git checkout FETCH_HEAD and git log (or whatever), view the diffs using git diff FETCH_HEAD (against your current branch) or finally integrate the changes he's asking you to using git merge FETCH_HEAD. Once you do this, you can push the changes to your own repository using git push origin master (assuming the original repository is added as origin).