You can geometrically show why it's like that for $S^2$, as every curve based in a point $x$ can be retracted to that point $x$, so the fundamental group is trivial hence equal to the point $0$.
But why is it true for $S^3, S^4, ...$ too?
It would be great to find an intuitive, geometrical way to explain it to others. I don't need a formal mathematical proof, but just the intuition behind it and why it makes geometrically sense.
Convince yourself inductively along the following lines. If I have a loop in $\mathbb{S}^{n}$, find a homotopy which contracts the loop to the equatorial sphere of a dimension lower. If the loop has become an interval, you're done, because this is contractible. If not, you can now continue the induction, forgetting the rest of the sphere and thinking now only within $\mathbb{S}^{n-1}$.