The most important reason for British involvement in Sudan was the Suez Canal.
The Suez Canal was completed in 1869. The canal gave Britain better access to India at a time when trade with India was vital for Britain. Because of this, the British soon came to regard the Suez Canal as a major strategic asset.
Since the Suez Canal was important, Sudan was now important. This was because Sudan borders on the Red Sea, which is at one end of the canal. Because of this, the British started to concern themselves with Sudan. They intervened in Sudan when there was a rebellion there in the 1880s and again when they feared French incursion in the 1890s.
All of this was done because of the strategic location of Sudan on the Red Sea.