捧げよ is a forceful, formal imperative meaning “dedicate it,” “devote it,” or “offer it up.” It is the commanding form of 捧げる, and its old-fashioned tone makes it especially dramatic and memorable.
In the song’s iconic rallying cry, 心臓を捧げよ！ means “Dedicate your hearts!” or, more literally, “Offer up your hearts!” It expresses total devotion and sacrifice for humanity’s future.
This is the opening theme of the Attack on Titan anime, and it sounds like a commander yelling at troops before a suicide charge. The singer opens on a bitter note: "これ以上の地獄はないだろうと信じたかった" (I wanted to believe there could be no hell worse than this). Then disaster arrives like a rude visitor: "扉を叩く音は絶えず酷く無作法で" (the knocking on the door is endless and terribly rude). I love that image. The end of the world is treated as bad manners.
The enemy is named bluntly: "奴らは駆逐すべき敵だ" (they are enemies we must wipe out). 駆逐 (kuchiku) means to drive out or exterminate, a very strong word you will hear all over this show. Then comes the famous chant, "捧げよ! 捧げよ! 心臓を捧げよ!" (Offer it! Offer it! Offer up your hearts!). It is the salute of the Survey Corps, the soldiers in the story who go outside the walls to fight the Titans. The 捧げよ form is an old-fashioned, formal command, which is why it sounds like an oath. The song closes on "進むべき未来を その手で切り拓け!" (cut open the future you must walk, with your own hands!). It is completely over the top, and that is the point.