[verb て-form] + ごらん [verb て-form] + ごらんなさい
てごらん gently tells someone to try something: the て-form of a verb plus ごらん. It is said from above – a parent to a child, a teacher to a pupil – and it comes with encouragement.
These go beyond plain てごらん.
ごらん is short for ごらんなさい. The full form sounds more like an instruction, from a teacher or a parent.
The word is ごらんご覧, the honorific for looking, as in ごらんになるご覧になる and ごらんご覧ください, "please look". After a て-form it keeps none of that respect. On its own, ごらん is a gentle "look!".