why did the U.K. have the British empire?

Because we wanted one!

In a way, it just happened. As an island nation, Britain depended on trade. Despite what many like to think, there was no vast army conquering countries. However, the Royal Navy was vast. It could take on the next two largest navies in the world at the same time.

to make money grow fat and be lazy and have thousands of servants it's own people never enjoyed the benefits of that fabulous wealth all over the world were giant buildings filled with servants the rich had a wonderful life but the poor were dogsbodies my ancestors were poor and live in the Squalor of workhouses this is their story you do not see a British writer describes it his name Charles Dickens

Mainly to establish trading monopolies.

Why does any empire have an empire? A country expands, conquers, benefits, expands further, and, just like the Roman empire, or the US empire lately, most of those won over enjoy the benefits of the conquest, so it feels right.