Download the plugin archive and expand it (you've likely already done this).
Upload the popularity-contest.php file to your wp-content/plugins directory (not in a sub-folder).
Go to the Plugins page in your WordPress Administration area and click 'Activate' for Popularity Contest. This will create the database table used by Popularity Contest.
Congratulations, you've just installed Popularity Contest and it is now tracking data for you.
Optional: go into Options > Popularity to modify the values of each view and feedback type.
Optional: set up a CRON job to run the 'reset feedback count' script every night (see how to do this in the FAQ section).