A strange person came up to me and asked for money.
Literal
Strange person [subject-が] approached come and money [object-を] give [quotative-と] said.
近づいて来て chains two actions: 'approached and came.' くれ is the blunt imperative of くれる — the stranger's actual demand was お金をくれ ('give me money'). The quotative と marks this as what they said. A vivid street-encounter sentence.