What is the etymology of using the word "out" after "thaw"? For instance, "I put the food on the counter to thaw out." Are there any other such words that are customarily followed by "out" where "out" could be omitted and not change the meaning?
Thawing is something that can be done partially, since it takes time and heat energy, and those are continuously variable, so thaw doesn't necessarily mean 'become completely unfrozen'. Therefore, a verb that does mean that is useful, and that's what phrasal verbs are good for.
As a particle on phrasal verbs, out generally adds a completive sense to whatever the verb is. A fire burns out or goes out when there is no more fire; a frozen article thaws out when there is no more frozen part.