It is well known that the operations of differentiation and integration are reduced to multiplication and division after being transformed by an integral transform (like e.g. Fourier or Laplace Transforms).

My question: Is there any intuition why this is so? It can be proved, ok - but can somebody please explain the big picture (please not too technical - I might need another intuition to understand that one then, too ;-)

The Fourier and Laplace transforms are defined by testing the given function f by special functions (characters in the case of Fourier, exponentials in the case of Laplace).

These special functions happen to be eigenfunctions of translation: if one translates a character or an exponential, one gets a scalar multiple of that character or an exponential.

As a consequence, the Fourier or Laplace transforms diagonalise the translation operation (formally, at least).

Whenever two linear operations commute, they are simultaneously diagonalisable (in principle, at least).  As such, one expects the Fourier or Laplace transforms to also diagonalise other linear, translation-invariant operations.

Differentiation and integration are linear, translation-invariant operations.  This is why they are diagonalised by the Fourier and Laplace transforms.

Diagonalisation is an extremely useful tool; it reduces the non-abelian world of operators and matrices to the abelian world of scalars.
