"Blessed are the pure in heart, for they will see God." (Matthew 5:8) (NRSV)
Psalm 24:3-4 probably serves as background to the fifth beatitude. The psalmist asks who shall ascend God's hill and stand in "his holy place?" The answer is he (or she) with clean hands, a pure heart, a life given to truthfulness. The Psalm may well have been used in worship, perhaps just before the people entered the Temple.
In any case, Jesus opts for a briefer statement. Who will see God? The pure in heart. The "heart," in the New Testament, is the will. We might paraphrase the beatitude: "Blessed are the pure in will, for they will see God." Jesus, to borrow a phrase from Kirkegaard, calls his followers to choose to "will one thing." They are to cultivate a single-minded devotion to God. Such focus may open our eyes, so that we see God.
The beatitude, of course, implies most of us lack such a focus. I think this is true. Most people of faith genuinely want to see (experience, know, etc.)God, but they also want a good many other things. To borrow from other sayings of Jesus, we find it hard to sell all we have in order to purchase the pearl of great price. We like to think we can keep what we have yet get the pearl as well. We're like a young man who greatly desires to marry a certain woman yet refuses to stop dating other women. A divided heart (will) ultimately leaves us out in the cold.
Perhaps this is why Jesus found it necessary to say, "Seek you first the kingdom of God, and all these things shall be added unto you."
What happens if we become "pure of heart?" At the very least, we start to "see God." We see what we learn to see. Focused on God, we begin to see the divine in others. Some of us catch a glimpse of God in history, a good story, or nature itself. The longer we focus on God, the more apt we are to see God in ourselves as well. As a teenager once put it: "Every thing in life becomes a God thing."
Most of us, myself included, never achieve sustained purity of heart. Fed by the occasional experience, however, our desire to do so grows. With practice, we get better. A kind of spiritual muscle memory develops. Good orship, both private and corporate, is largely devoted to strenthening our spiritual muscle memory.