When someone compares the density of water to the density of the air and steel of the boat, the boat is much lighter, allowing it to float. Before water displaces the boat, only a very small part of the boat has to submerge into the water.
The fact that the boat floats is due to Archimedes' principle. With this principle, when an object pushes water away, the water puts force back onto it equal to the weight of the object. This allows the object to float. This is also what happens when someone sits in the bathtub and, as they sit, the water gets pushed away.