Since God is so present and intimate with us in our prayer closets, you can see how inappropriate it would be to speak to him in "meaningless repetition" as if he doesn't hear us. Some scholars think Jesus is referring to the Gentiles' prayers of incantation during their pagan magical rites. I've noticed that most non-Christian religions teach people to relate to the gods or spirits with a great deal of anxiety and insecurity. The gods must constantly be called down from their own ethereal realm and pleaded with, using works or rituals or sacrifice or chanting as leverage. It's up to the suppliant to jump through the necessary hoops to catch the god's or spirit's attention and persuade it to do what he asks.
Unfortunately, I have to admit that I've mainly experienced this type of anxiety and insecurity while worshipping the true God, particularly during some of the worship services and retreats I used to go to in my college days. I used to get caught up in some seemingly powerful worship services, where the music and repetitive singing made me feel like I was trying to drive my worship up to heaven by the sheer force of emotional intensity. Yet once the music stopped, I used to wonder why (though it was hard to admit to myself at the time) I was left feeling even more spiritually distant from God than before. The reason may lie with what Jesus says: it was my many words that left me feeling empty.
Someone once pointed out to me that our worship songs are simply prayers put to music. When viewed from that perspective, and in view of Jesus' teaching here, it's clear that the lyrics of some of our songs need to be reworked to express more clearly and rationally what we want to say to God, instead of devolving into the kind of mindless repetition that Jesus finds so abhorrent. Otherwise you are left feeling empty. You didn't really commune with God. You didn't speak to him simply, directly or honestly from the heart. Your many words and repetitions actually estranged you from him instead of drawing you into greater intimacy.
At root, I think the entire premise behind it all is the false notion that we are pursuing God and trying to woo him into accepting and noticing us. Coming to him with prayer chants is treating him as if he is running away. But the truth is that we don't pursue God at all. We have never pursued God; he's the one who pursues us. He's the one who has come down from heaven and become a man to walk in our shoes. He is the one who seeks to make our very bodies the temple of his Holy Spirit. But we respond by throwing pretend prayers at him, and when he doesn't answer we say that we tried to pray but it didn't work, and that gives us an excuse to go off and do what we want. I think Jesus is saying we have to stop playing games and realize we are talking to the God who is, in fact, so close he knows what we need before we even ask. And next he wants to instruct us through the Lord's Prayer on how to address our very imminent and present heavenly Father.