Coefficients of the complex Fourier series give spikes at discrete frequencies. I'd like to understand why F.T. gives us the continuous frequency spectrum?

That is the case because the real line is not compact.

The Pontryagin dual group of a locally compact abelian group $G$ is discrete (or has any isolated point) if and only if $G$ is compact. But the circle group giving the Fourier series is compact. It is an essential point that the characters of a non-compact lca group are not square-integrable (you can easily prove that). If there would be an isolated point $x$ in the dual space, then the characteristic function $\chi_{\left\{x\right\}}$ (in your language that is a “spike at a discrete frequency”) would be non-zero in the $L^2$-space. The inverse of the Fourier transform would have to map this non-zero $L^2$-function to a non-zero $L^2$-function, which would simply be a character $\exp(ikx)$, but this character is not square-integrable.

Actually the characterisation by square-integrability of the characters/representations is more fundamental than the characterisation using compactness: In the non-abelian case an irreducible unitary representation is a discrete series representation (i. e. it contributes to the inverse of the Fourier transform as a simple summand, or ”spike”) if and only if there is any square-integrable matrix coefficient. But the group might be non-compact