架子 = 'rack / frame' (same word as in 摆架子 'put on airs'), 鼓 = 'drum'. Literally 'drums on a rack' — a nicely descriptive name for the Western drum kit, distinguishing it from traditional single Chinese drums like 腰鼓 or 锣鼓.
usage
The verb for playing drums is 打 (dǎ, to hit), not 弹 (tán, to pluck) — so 打架子鼓 ('to play drums'), same pattern as 打鼓 for any drum.
