sequences and series - Proving the identity $sum_{k=1}^n {k^3} = big(sum_{k=1}^n kbig)^2$ without induction

I recently proved that

$$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$

using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial interpretation of this property. I would also like to see any other proofs.

Answer

Stare at the following image, taken from this MO answer, long enough: