The British did own the island of Hong Kong. What they leased was the Kowloon peninsula with New Territories, Kowloon and some other important areas.

So Hong Kong, Stonecutters island, and parts of the Kowloon peninsula were give to Britain under the Convention of Beijing in 1860. Later Britain leased Lantau, New Territories and a bunch of islands. For 99 years.

Well during the interim Hong Kong expanded. Eventually more resources were needed. Mainly water. Hong Kong negotiated with China to get more water.

Eventually as the lease conclusion started to come closer Britain realized China wouldn't renegotiate the lease. They could either give up the leased territory and have a huge population crisis, try to fight China over it and lose water to the whole area, or negotiate a third option.

Simply, it became infeasible to keep Hong Kong as China's strength grew. They also didn't want to go back on an international agreement and wanted to increase relations with China.