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After the death of Sultan Khalid, the British supported the appointment of Sultan Hamad bin Thuwaini as his successor. However, Khalid's cousin, Sultan Khalid bin Barghash, declared himself the new sultan and refused to step down. The British, having significant military presence in the region, issued an ultimatum demanding Khalid's surrender by 9:00 a.m. on August 27, 1896.
When the deadline passed, the British launched a bombardment on the Sultan's palace and other strategic targets using warships anchored in the harbor. Zanzibar's defenses, consisting mainly of outdated wooden warships and a few shore batteries, were no match for the superior firepower of the British.
Here's the Wikipedia page about the war. Interesting read.