Question

Why is Gulliver's so eager to make his country appear very important to brobdingnagians

Answer 1

due to some problem in his country

Answer 2

Gulliver wants the brobdingnagians to think that his country is important. The Lilliputians also want Gulliver to think that their country is important. The connection is that both are wants to think that their country is important. They are intimidated to some extend by physical size and they equate size with importance. So they want to prove to the larger people that they are important. Its like little brother wants the bigger brother.