On the surface, the verbs to leave and to let might seem like quite distinct entities. To leave means to go away, and to let means to give permission. Simple.
But the words are closer than you might think. If someone is annoying you, they might be told to leave you alone. Or, to let you be. If you want to take a break for a while from your job, you have to ask for permission. And if that information is granted, you get a leave of absence. Sailors who get to take a break go on shore leave. You could look at that in two ways: they leave the ship to go on shore, or they get leave to go on shore.
As you can see, the two words are a little more interconnected than they might appear. They don’t share the exact same etymology, but they do separately come from two different, but very similar Latin words: to let from lætan, and to leave from læfan. I wouldn’t be surprised if the similarity between the two verbs led people to confuse the two. And even if we just think of to leave in terms of going away from somewhere, the meaning isn’t hugely different from to let. If you let something remain or let something be, you leave it alone by staying away from it. You quite literally are leaving it alone.
It’s no wonder that the two can be used in phrases which have identical meanings. Even today, in different dialects of English, the two are interchangeable. In some parts of the world it’s not unusual to hear someone say Leave it be! or Let it alone! And even if you’ve never heard of those before, they might not sound so strange, because we’re so used to so many similar expressions which use one or the other word. It’s not also not unheard of for English learners to confuse the two. French, for example, uses the same verb for both (laisser), and French speakers can often mix the two up.
Still, the sound wouldn’t be quite so good if it was called Leave it Be.