The heart is what you are, in the secrecy of your thought and feeling, when nobody knows but God. And what you are at the invisible root matters as much to God as what you are at the visible branch. "Man looks on the outward appearance, but the Lord looks on the heart" (1 Samuel 16:7). From the heart are all the issues of life.
What comes out of the mouth proceeds from the heart . . . For out of the heart come evil thoughts, murder, adultery, fornication, theft, false witness, slander. These are what defile a man. (Matthew 15:18–19)
Either make the tree good, and its fruit good; or make the tree bad, and its fruit bad; for the tree is known by its fruit . . . For out of the abundance of the heart the mouth speaks. (Matthew 12:33–34). So the heart is utterly crucial to Jesus. What we are in the deep, private recesses of our lives is what he cares about most. Jesus did not come into the world simply because we have some bad habits that need to be broken. He came into the world because we have such dirty hearts that need to be purified.
So what is it to have a pure heart?
Who shall ascend the hill of the Lord?
And who shall stand in his holy place?
He who has clean hands and a pure heart,
who does not lift up his soul to what is false,
and does not swear deceitfully.
You can see what David means by a "pure heart" in the phrases that follow it. A pure heart is a heart that has nothing to do with falsehood. It is painstakingly truthful and free from deceitfulness. Deceit is what you do when you will two things, not one thing. You will to do one thing and you will that people think you are doing another. You will to feel one thing and you will that people think you are feeling another. That is impurity of heart. Purity of heart is to will one thing, namely, to "seek the face of the Lord"
What is it to see God?
First, to see God means to be admitted to his presence.
Second, seeing God means being awestruck by his glory—by a direct experience of his holiness.
Finally, seeing God means being comforted by his grace.
So when Jesus promises the reward of "seeing God" there are at least these three things implied: we will be admitted to his presence, not just kept in the waiting room. We will be awestruck with a direct experience of his glory. And we will be helped and comforted by his grace.
And this we will have—in part now, and fully in the age to come—if we are pure in heart.
Excerpts taken from John Piper, Blessed are the pure in heart