Let's look at the immediate context:
“And when you pray, you must not be like the hypocrites. For they love to stand and pray in the synagogues and at the street corners, that they may be seen by others. Truly, I say to you, they have received their reward. But when you pray, go into your room and shut the door and pray to your Father who is in secret. And your Father who sees in secret will reward you.
“And when you pray, do not heap up empty phrases as the Gentiles do, for they think that they will be heard for their many words. Do not be like them, for your Father knows what you need before you ask him. Pray then like this:
“Our Father in heaven,
hallowed be your name...—Matthew 6:5-9 (ESV)
This is the Sermon on the Mount, which might be summarized as calling Jesus' disciples to go beyond the normal or surface understanding of obedience. The first paragraph I quoted above, for instance, calls us not just to pray, but to pray in secret when it produced no earthly benefit. We aren't supposed to make a show of prayer, but to pray intimately with our Father.
Similarly, Jesus prefaces the actual text of the "Lord's Prayer" with the warning not to use empty phrases. Don't approach prayer as a spaghetti test. Our words must be meaningful and not merely repetition of a formula.
Although we now think of the "Lord's Prayer" as a very standard way to approach God, it was probably as strange to the first audience's ears as the rest of Jesus' teaching. The NET Bible notes:
Pray this way. What follows, although traditionally known as the Lord’s prayer, is really the disciples’ prayer. It represents how they are to approach God, by acknowledging his uniqueness and their need for his provision and protection.
God is addressed in terms of intimacy (Father). The original Semitic term here was probably Abba. The term is a little unusual in a personal prayer, especially as it lacks qualification. It is not the exact equivalent of “daddy” (as is sometimes popularly suggested), but it does suggest a close, familial relationship.
So it seems natural that Jesus was calling us to something beyond mere obedience to pray these particular words. Rather, He was calling us to pray with a particular attitude. Rather than obey a distant deity who has no particular reason to listen, Jesus suggests we treat God as a close relative who cares for us.
Personally, I find the prayer very carefully structured and well suited to be a guide. I sometimes pray one phrase at a time and pause to meditate on what I personally wish to communicate along those lines to my God. As long as the Christian is thinking about the meaning and not merely repeating the sounds, there's nothing wrong with reciting the prayer word for word, however.