The Jesus prayer is a very short phrase: "Lord Jesus Christ, son of God, have mercy on me, a sinner." It springs from the word of Jesus that we have in the Gospel of Saint John, where Jesus in his last discourse to his disciples says, "You have never asked anything in my name. Now, anything you ask in my name will be given to you." The idea of asking in the name of someone is something we're not too accustomed to these days. We think instead of back door politics: Knock, knock, knock. "Who's there?" "George sent me." The door opens and out comes the little money bag and off we go.
In the Near East of Biblical times, "name" meant the presence or reality of the one whose name was called. That's part of the reason why the name of the God of Israel became unspeakable: the name was never adequate to the reality. So asking in Jesus' name is making present the full reality of what Jesus is, which is being present immediately to God.
This presence is not a confrontational one. It's not the presence of speaking with someone on the phone. It is an immediate and absolute union, like the presence of two people in love: not something you intellectualize, not even necessarily emotional. It's just there.
One of the best examples: two people who've been married a long time and have been through the good times and the bad together. One can be in the kitchen and the other in the living room, but they're completely aware. Or one is doing a crossword puzzle and the other writing a letter, but they're absolutely present to each other.
The Jesus prayer is a vehicle to achieving that presence with God. Using words makes it easier for us, just as between two people who love each other a glance or kiss makes it happen.