The heart, in the classical liberal anthropology of the human person, is the symbolic seat of the Will–the power of decision and freedom. It is important to realize that this has changed in modern times; where the heart is now seen as the seat of the Emotions or even the Senses, which were previously seated in the viscera and the loins, respectively. When the Scripture refers to the “heart,” then, it is often referring, metaphorically, to the will (not to the emotions as is often assumed… for instance, in the post above me).
The will is ordered towards good. That’s what it’s pointed at. It aims at happiness; it aims at the good. The ultimate good is God, so a properly functioning will also points unerringly at God. That’s a fact of the human person… at least, it is when we’re working properly.
The Christian understanding of the Fall is that original sin damaged the Intellect and turned the Will away from God. We are still aimed at the good, but our intellect often errs at perceiving what is truly good, and our will often rebels against pursuing the higher goods. The will, rather than aiming at the final good, which is God, was turned away and disoriented, so now it perceives lesser goods to be greater than God and so instead pursues those. Thus, we say (somewhat metaphorically) that the Fall “darkened our intellect and hardened our hearts.”
When the Bible says to not harden your heart, it is reminding you that your will, as a result of the Fall, is in rebellion against God. You need to tame it by practicing virtue. Otherwise, you will not be free to follow God when called, you’ll turn your back on HIM, and you will ultimately lead yourself to destruction. Your heart hardens into sin; you have to violently wrench it back onto a right course.
At least, that’s how I was taught. The passage is definitely talking about the will and not emotions, inclinations, or feelings… but there’s no doubt a lot of room for interpretation within that.