Oskar is such a wonderful character, and it is because of his ability to capture a truth of life or an emotion with a childish metaphor that, in actuality, is richer and more original than the things most adults would come up with. He mentions his "heavy boots" in an almost off-handed way very early in the novel and the reader is a bit perplexed, but his metaphor becomes clear as the novel progresses. The heavy boots are any and all of the facts of his life that weigh him down. If you think of boots as a metaphor then you think of the following: boots protect your feet, boots allow you to move around in rough terrain, boots are more substantial than other footwear. But, if your boots are heavy, then your protection is perhaps "too much" or burdensome and your movement and forward momentum is potentially impeded by the weight of them. The list of events and facts that make Oskar's life-journey burdensome come from many things: most notably the death of his father in the 9-11 attacks, but also the hardships of his search for the lock that matches the key he found, his mother's new relationship, his disconnection from classmates, his carrying the burden of having heard his father's last messages on the answering machine, etc. Oskar is bothered by the fact that the family buried an empty casket. Oskar is troubled by some of the circumstances of the people he meets -- especially the "boarder" who lives with his grandmother and lonely Mr. Black who lives upstairs.
Oskar is an incredibily sensitive little man who is over-burdened by the harsh realities of life, and therefore, must learn to either take off the heavy boots, adapt to the heavy boots, or lighten the emotional load he carries. By the end of the novel, there seems to be some hope that his boots will be a little lighter: he solves the mystery of the key and better understands the people around him.