Expect a buoyancy force exerted on something when that thing makes a hole in a fluid, replacing the gas or liquid that would otherwise be there. Re-imagine the thing extracted from its environment by choosing to replace the surrounding liquid or gas by a compression force exerted on the thing.
The larger the hole you make, the greater the buoyancy force.
Fluids always have particles moving around in them. Push something into a fluid, and the particles bouncing off the surfaces are the mechanism for the buoyancy force.
You're going to think about making holes in water, using boats. Making these holes results in a buoyancy force on the boat, which supports the boat and its cargo. Buoyancy forces only occur in fluids, so you'll be thinking about pressure and the particles of the fluid bombarding the surfaces of the boat. To think about bombardment on surfaces, we'll choose to make a simple boat. You'll be concentrating on the surfaces at the sides and the bottom.
Here is how. Start with a boat floating on water.
Make the boat simpler, to help your reasoning.
In a fluid, pressure increases with depth. So the sides of the boat will be at the same pressure, but the top and bottom will be at different pressures.
As the pressure increases with depth, so the bombardment gets more intense. Therefore you might expect different surfaces of the boat, at different depths, to experience different bombardments on each piece of the exposed area.
Focus on a surface, and vary the pressure on either side. Increasing the pressure increases the bombardment. A difference in bombardment results in a force.
Think about both the bottom surface and the two surfaces that make up the sides of the boat.
You get a single buoyancy force as a result of all those bombardments. A forces view is simpler.