If you understand fourier transform well, then perhaps this viewpoint will help. The fourier transform is 'just' a change of basis, with the basis being the sinusoidal functions. Why these? Well, because if we take a look at the discrete fourier transform, the matrix that changes the basis is both unitary [which is has to be as a change of basis] and vandermonde. So we can think of it as both a 'change of basis' and as a 'evaluation of polynomial'. This is where most of the power of the fourier transform comes from.

Similarly, the laplace transform is also a change of basis. But the basis it chooses is a very special one --- it's the eigenvectors of the differential operator. Note that

    d/dx(e^(ax)) = ae^ax

So e^(ax) literally an eigenvector of `(d/dx)`. And as we all know, going to the eigenbasis of a given operator/linear transform/matrix makes it easier to manipulate. The laplace transform is a change of basis that digonalizes the differential operator. This makes it easy to solve differentials.