To empty your mind means to let go of your preconceptions.
This is important because that way you will not favor any particular facts and get distracted by them.
I believe that the reason why we all recognize some kind of connection between different Zen koans is that the unenlightened ask questions based on wrong assumptions because they have not emptied their minds and therefore have a skewed perspective.

A popular word describing this proces is to unlearn.
You have to untravel the roads you have traveled before you can go any further.
Or, more precisely, you have to backtrack and go in another direction.
This doesn't mean that the information you learned when you went in the other direction is useless in general, it just means that it's useless and even dangerous where you're going.

The simplest example I can think of involves complex numbers - all through school you have learned what you can and can't do with numbers and when you learn complex notation it's suddenly a whole new can of worms.
The operations become nontrivial once again and the things you can and can't do are redefined.
Instead of a superset of real numbers you can instead view complex numbers as a new algebra (which is good, since that's what it is) and reject the rules of real numbers altogether.
This prevents you from making mistakes based on rules learned about real notation.

Most of what you learn will at one point make you say "Is that really it? It's far simpler than I thought." and the reason for this is that you had an assumption about how things worked when in fact that assumption was useless in this context. In fact, you have just acquired some gumption in the subject.