Why does FFT generate a jaggedy signal
What you are seeing is a peak from sinusoid convolved with a spectrum of rectangular window sinc function. That is due to property:

$$x(t)\star w(t) = X(f)\cdot W(f) $$

Where your signal $x(t)$ is a sinusoid with a spectrum being a Kronecker (Dirac) Delta function, $w(t)$ is a rectangular window function. In DFT by default we are using rectangular window (taking a time snippet of a signal). Please check out this paper for more info: A Comprehensive Window Tutorial