Countable compactness and locally finite family of subsets.

Question

The title pretty much says it all, but I admit that I got a bit confused when trying to do the proof of the following result: if $X$ is a countably compact space, then every locally finite family of nonempty subsets is finite. I have tried following the steps in the proof of similar result for compact spaces (i.e. the one provided here https://math.stackexchange.com/q/2579547), however... I find it hard to create a countable cover of $X$ in the similar manner it is done in the original proof. Could anyone provide me a tip on how to proceed with the proof?