Answer:
Bert swallowed the worm with a grimace
An action verb is a verb which describes an action. Let's take a look at all the sentences, and look for a word which describes an action:
1) Leslie was the best swimmer in the group
in this sentence there is no action verb, swimmer might seem like it, but it's not an action, it's someone
2) I am usually the quietest person in my class
This also does not contain any action. Quietest is an adjective, describing how something is
3) My cat is always on the windowsill
this also contains no action
4) Bert swallowed the worm with a grimace:
This sentence does contain an action, swallowing. An action means to do something, in this sentence Bert is doing an action, swallowing. Grimace is not a verb, it's showing how he did it, with an adjective
Answer from Gauthmath