Learn English – What does it mean by “higher register”

As others have said, it's not a standard phrase. However, it's important to point out how the word heavy can be used to describe things other than weight and mass. Couples can engage in heavy conversations; students can trudge through heavy reading passages. We can be advised not to eat a heavy meal before a long swim.

Your passage doesn't even tell us what is being described – something you should be more careful about in future questions. However, I can venture guess as to what the expression might mean, based on Definition 6 in Macmillan, which says:

heavy (adj.) used about things that look ugly because they are big
heavy features (=large mouth, eyes, and nose) : He was a tall dark man, with heavy features.

to harness means to make use of something that's typically very hard to make use of (our civilization, for example, has harnessed the electrical phenomena to make our lives better). The idea here is that the way the human mind works is still poorly understood even though scientists have been studying it for centuries now trying to understand how it really works.

So, at this point in time, the human mind is still a mystery to us, but we do know, though very little, something about it. For example, we all know that rote repetition aids retention—the longer you repeat or are exposed to something, the easier it becomes to retain it in your memory. That's one fact.

The other fact about the mind is that when sometimes you hear one of those catchy tunes, the lyrics tend to stick momentarily even without long exposure to the song. We all know that happens. So, what they're talking about here is that we can take advantage of this phenomenon and put that to good use—learning foreign languages, for example. Although we still don't understand how the mind works, this method offers us a small window into the human mind so that we can try and improve our ability to learn vocabulary and phrases as well as retain all that information.