Learn Turkish: Müşteri tutup garsonu arsızca tersledi. - The customer went and shamelessly snapped at the waiter.

No, there is no physical grabbing happening. In this context, tutup (just like kalkıp) is used as a converb to show that the following action is unexpected, inappropriate, or audacious. It acts just like the English phrasing "went and...", as in "The customer went and snapped at the waiter."

English speakers say "snapped at the waiter". Why isn't it garsona (dative) instead of garsonu (accusative)?

Because verbs of communication and emotion don't always map perfectly between the two languages. While English uses the preposition "at", the Turkish verb terslemek takes a direct object. Since it's a specific waiter being snapped at, it takes the accusative case suffix -u, becoming garsonu.

How is the word arsızca built?

It is built from the root word ar, which means "shame" or "modesty". Adding the privative suffix -sız (meaning "without") creates the adjective arsız (shameless or impudent). Finally, adding the adverbial suffix -ca turns it into the adverb arsızca (shamelessly).

What is the literal meaning behind the verb tersledi?

The root is the word ters, which generally means "opposite", "reverse", or "wrong". The verb form terslemek literally means to treat someone in a dismissive, harsh, or contrary manner. It is the standard Turkish equivalent for "to snap at" or "to rebuff" someone.