Can someone explain to the unaware person (i.e. me) in the room why one should care about this result?

It's a famous open problem largely because the bound R(k) < 4^k (roughly) was proven close to 100 years ago by a very simple induction, taught in any first course in discrete math. This is the first time it's been improved to (4 - c)^k.

No-one thinks that's actually the right answer, which is probably 2^{k/2} or thereabouts, so probably one day long in the future the result under discussion will be forgotten. But this is certainly a very big breakthrough in this area, to add to several other big breakthroughs in extremal comb and related areas such as Kelley Meka on progressions, or Gilmer on union-closed.

As far as I know the lower bound poses a more painful problem to experts in the field. A simple probabilistic construction by Erdos from 1940s, improved by a factor of $2$ by Spencer in 1970s using Lovasz's Local Lemma and that's about it. No explicit construction giving exponential lower bounds even.

It seems like the off-diagonal Ramsey problem was also mentioned in the talk. Did the same group of researchers also make significant progress on this problem as well?