What do we mean when we say it today?
Shmoop's heart is content when people read Shakespeare. We love it. And that's all this phrase means. Something is stellar and you're happy about it. People use it when it's true most of the time. Henry VI might have started the trend, but we keep it going.
Here's the thing, though: we use this phrase a bit differently today. We often use it at the end of a phrase to indicate that we do something a lot. As in, she ate cookies to her heart's content (meaning she ate and ate and ate until she could eat no more).