Option A is the answer.
The sentence defines a time in the past (the time at which I arrived home). That time was the start of the action of David's playing. The adverb "since" also means that the action is still occurring. So present perfect (A) is the best option for the verb tense.
If I change the adverb:
When I got home, David ___ online games.
the answer is (C) - was playing.
And if I change the tense in the clause:
When I get home, David ___ online games.
The answer is (D) - will play