IIRC, this was already the standard story of snake evolution. I believe the remaining mystery revolves around which lineage they evolved from. Skinks, monitors and I believe Plesiosaurs are proposed as possibilities...

At present, there are multiple lineages of non-snake, legless lizards. That basic body plan is relatively abundant, probably was then too. One of these lineages started to diversify into snakes after KT. Large fauna extinctions probably made small prey more abundant, so it's a good time to be a snake. A lot of snakes are mammal specialists, which also succeeded after KT. Snakes may have started to proliferate in response to the post KT abundance small burrowing mammals.