What is the physics behind the width of a main lobe?
A non-rectangular bell-shaped window reduces information magnitude at the edges of the window.  This either is, or acts like, information loss.  Thus the main lobe gets fatter to indicate a greater uncertainty with respect to its peak due to information loss. (relative to the noise floor, etc. etc.)

The concepts of information theory seem to be used in both physics and signal analysis.

The side lobes get lower because the high frequency energy related to producing any discontinuities between the FFT's 2 sides using circular basis vectors is reduced by a bell shaped window.

The two effects are similarly opposite in term of energy trade-off, but not exactly opposite, as different windows have differing areas under the curve, thus producing another degree-of-freedom in the trade-off between main lobe and side lobe energy, even given Parseval's theorem.

The width of the main lobe is inversely related to the length of the information and S/N of the information fed to the FFT.  In absolutely zero noise, just 3 or 4 non-aliased points can be used to calculate the exact frequency of a single pure sinusoid in the time domain (e.g. equivalent to an interpolated main lobe width of zero in the frequency domain).  Additional information outside the DFT data can also be used to estimate a narrower main lobe: For instance, many FFT post-processing frequency estimation methods assume a-priori that the data is stationary outside the widow and thus make use of the results from other offset windows, etc.