Voting on these sites is usually straightforward, you load the page by clicking one of the servers at the top of the page.
Log into the site if required and find the "Vote" or "Steam Vote" button, sign in with steam if asked and then you should see a success / confirmation message that you have voted.
If after you have logged into steam and you have not been returned to the vote site, start the process a second time and you should be successful the second time.
How do you claim rewards?
After you have voted, in game run..
In Game Chat Command: /claim
If the rewards are not rewarded immediately, just give it 5 minutes or so as the Vote Sites may be experiencing a busy period.