、。
As soon as he got off the train, he was met by his cousin.
Literal
Train [object-を] got-off as-soon-as, he [topic-は] cousin [of-の] greeting [object-を] received.
~とすぐに ('as soon as ~') attaches to the dictionary form of a verb to express immediate succession. 出迎え is the nominalized form of 出迎える ('to go out to meet') — meeting someone on arrival, typically at a station or airport. 出迎えを受ける is the standard collocation ('to receive a greeting/meeting'). The から ablative is replaced here by を — 列車を降りる uses を on the thing you dismount from, not から.