What was Sir Humphrey Gillbert's reason for exploring?

Answer 1

He was certainly hoping to make money.

But the fact that he was stepson to Walter Raleigh's father, grew up with the (to be) Sir Walter Raleigh, and remained his close associate throughout his life, presumably sparked Gilbert's interest in exploration and colonization.

In Queen Elizabeth's England, there was something akin to an informal "Explorers Club" actively swapping ideas and getting one another involved in planned expeditions (many of which were abortive). In addition to Raleigh, Gilbert was involved with at least two other famous explorers, Martin Frobisher and John Davys.

Answer 2

Essentially he was trying to make money. Either directly through schemes of colonization or by gaining royal favor.