I don't like sending postcards while traveling.

Literal

Trip-during [time-に] postcards [object-を] sending-out [topic-は] not-liked.

葉書を出す (to send a postcard) uses 出す in the sense of 'to send out/mail.' ~のは好きではない (don't like doing X) nominalizes the action with の and negates 好き. A personal preference stated in plain past negative.