When I got off the train, it started snowing heavily.
Literal
Train [from-から] got-off time [at-に], heavily snow [subject-が] began-to-fall.
列車から降りる is slightly unusual — the common pattern is 列車を降りる, with を marking the train as the point being left. Using から is also grammatical and emphasizes the ablative 'from' sense. 降り始めた combines 降る ('to fall' for rain/snow) + 始める ('to begin') as a compound verb.