How does moderating more accelerate the reaction?

The fission events produce a high proportion of fast neutrons. The fission cross-section, ie chance of inducing a fission event when struck by a neutron, depends on the speed of the incoming neutron, and is smaller for the faster neutrons than for slower neutrons.

Thus by slowing the fast neutrons down there are more neutrons with a greater chance of causing a fission event, hence stronger chain reaction.

More neutrons will be able to interact with the fissile nuclei rather than just fly out of the reactor, so you get more fission events.

It would be more accurate to say that, at 10+ second time scales, nuclear can have some convenient load following characteristics. But it takes a while for any load change to feed back through the turbine, then the condenser, then the steam generator, then the primary coolant loop, then the reactor, then ...

And the economics of nuclear - enormous fixed costs, minimal fuel costs - encourages utilities to run them near full power. So they'll have relatively little spare generating capacity, regardless of their load following characteristics.

OTOH, a gas turbine generating plant is essentially a turboprop airplane engine, with the propeller replaced by a large generator. 1950's mechanical control technology can adjust the throttle on a ~1-second time scale as the load on the grid changes.

And the economics of gas turbines - quick & cheap to build, expensive to run - encourages utilities to run them (or not) as demand requires. So they'll usually have lot of spare generating capacity, to meet demand surges.

I believe only reactors used in the navy have this characteristic, the poison transients from adjusting load to quickly (and maybe having to shim for temperature) on a commerical reactor are liable to cause an unplanned shutdown.