Why are neutron absorption cross sections high at low incident energy?

This is because U-235 is fissile, that is you only have to deliver the neutron to the nucleus for the magic to happen. Unlike U-238 where just delivering it doesn't do the job, there you also have to impart the nucleus with the neutrons kinetic energy.

Once we know this, it becomes clear that for low energy neutrons, their de Broglie wavelength is very big. So the cross-section is effectively determined by the quantum size of the neutron, rather than any other dynamics, so roughly $\sigma \approx \pi \lambda_{dB}^2\approx \frac{1}{E}$