Does an increase of area lead to more collisons in a confined jar of gas?
Pressure is force over an area, or as an equation you might state that as:

$$P=\frac{F}{A}$$

Which can be rearranged as

$$F=P \cdot A$$

You are correct in saying that if the number of molecules are held constant, that the force would be over a larger area and therefore decreased. As you increase the area but keep the number of molecules constant, the pressure decreases proportionally, and no change in force is seen, but what Feynman was talking about was keeping the amount of molecules per cubic centimeter constant, or in other words keeping pressure constant. This way as area is increased, you see a proportional increase in force.