、。
I always start feeling sleepy after dinner.
Literal
Dinner-after, always sleep-[appearance-そう] become-[explanatory-んです].
眠りそうになる is the chain: 眠る (to sleep) → verb-stem 眠り → ~そう (imminence/appearance) → に + なる (become so). So literally 'become such that [I] look about to sleep.' ~そうになる is a useful productive pattern for 'start to become X' or 'feel about to X' — can describe emotions, physical states, or near-occurrences.