Any object will float on water if its gravitational force (downward force) is less than the upward force. The upward force is caused by the amount of water under the object. More there will be water under the object the better the object will float and will not drown.
Since a boat has an expanded size, its downward force is spread over a large area of water, and the water below is enough to give the upward force to keep the boat on the water surface. This explains why a huge boat will not drown whereas a small stone will drown. Furthermore, boats are designed specially in such a way that they would displace enough water to ensure they will float easily.
When weight is put on a boat, it is also spread in the boat, also on the water surface. If the weight did not spread, the boat will sink. Hence, boats are not only designed in such a way that these are expanded, these are only designed in such a way that when weight is put on them, it is spread across their body.
This concept that when the upward force is higher than the downward force is called Buoyancy.