Breakdown of Doch, bei diesem Brettspiel darf man zwei Karten ziehen.
zwei
two
dieser
this / that (fem.: diese, neut.: dieses)
dürfen
to be allowed to, may
bei
at, near, with
Questions & Answers about Doch, bei diesem Brettspiel darf man zwei Karten ziehen.
What kind of statement is the speaker replying to with Doch here?
Why is it diesem and not dieses Brettspiel?
The preposition bei always forces the following noun into the dative case. Since das Brettspiel is a neuter noun, the word for 'this' changes to the dative form diesem.
Could I use kann instead of darf here?
You could, and people would understand you, but darf (from dürfen) is much more precise here. Dürfen specifically means to be allowed to do something (having permission), which is exactly how board game rules work.
Why is the verb ziehen all the way at the end?
This is the standard rule when you use a modal verb like dürfen. The conjugated modal verb (darf) takes its normal position earlier in the sentence, and the main action verb (ziehen) gets pushed to the very end in its basic infinitive form.