、。
When they saw me, they suddenly stopped talking.
Literal
I [of-の] figure [object-を] see-[when-と], they [topic-は] suddenly talk [object-を] stopped.
姿 ('figure, form, appearance') is the visible form of a person — 僕の姿を見る literally 'see my figure' = 'see me.' The conjunction と here marks a 'when/whenever' — a conditional/temporal connection expressing immediate sequence. 急に ('suddenly') modifies the verb. The suggestive feel is that the people were gossiping about him.