Suppose f and g are normalized distributions and consider the function f(x)·g(y). If we "collapse" (integrate) in the y-dimension we recover f(x). If we collapse in the x-dimension we recover g(y).

But if we collapse along the lines x+y = v, we obtain the convolution f⋆g(v).

This picture is a little more advanced, but it makes clear two key properties of convolution: symmetry (commutativity) and the fact that the total integral of the convolution is the total integral of f(x)·g(y).