Maybe it would help your intuition to think about the discrete case first where the convolution is a sum rather than an integral. (f*g)(x) is the sum of f(i) g(j) over all (i, j) that sum to x.

Or maybe you could think of convolution as a kind of multiplication. Convolution makes certain function spaces into algebras.

Or you could think in terms of Fourier transforms: the Fourier transform of f*g is the product of the Fourier transforms of f and g.