This sentence uses the verb remercier which means "to thank". Here it is conjugated in the 3rd person plural → remercions. It is more expressive of gratitude than merci which is used as an interjection.
Would this be commonly used in casual speech, such as when a waitress has just brought food to our table? Or is it more common in written French than spoken?
To a waiter/waitress you would simply say "Merci" - it is an interjection. Remercier is the verb you would use to thank somebody for doing or for having done something.
Je te remercie d'être venu m'aider à déménager.
"Thank you for coming to help me move."
Doesn't it also mean to fire somebody? How can one distinguish the two meanings without context (like in this sentence)?
se faire remercier means to let go (a euphemism for dismissing someone)
vous here is a direct object and direct object pronouns usually come before their verbs.