WARNING: THIS USER SCRIPT NO LONGER WORKS - IT WAS ORIGINALLY MADE FOR THE TWITTER WEBAPP PRE JULY 2019
Download either Tampermonkey for Chrome, or Greasemonkey for Firefox if you want to run this script repeatably:
Tampermonkey: https://chrome.google.com/webstore/detail/tampermonkey/dhdgffkkebhmkfjojejmpbldmpobfkfo
Greasemonkey: https://addons.mozilla.org/nl/firefox/addon/greasemonkey/
Add the script via the dashboard of Tamper/Greasemonkey, and then simply navigate to https://twitter.com/i/likes to download your own likes, or replace 'i' with any other username to download their likes.