When should I say this phrase?
Bīru no o kawari onegaishimasu.
I couldn't figure out the meaning. My main source of confusion is that I've never seen this sentence construction: [noun]のおかわり.
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「おかわり」 means "another helping", "refill", etc.
So, 「ビールのおかわり」 just means "another beer".
Therefore, you say:
「[Name of food/drink item] + （を） + お願いします。」
Whenever you have finished your current serving and you are wanting to have another.