How did Egypt become a protectorate of the British empire?

As pointed out by another poster, Egypt came under "foreign" (i.e. French) influence in connection with the (French) building of the Suez Canal in 1869. This was true even though Egypt was technically a semiautonomous dependency of the Ottoman Empire.

Shortly afterward France lost the Franco- Prussian War of 1870-71. This led, in the late 19th and early 20th centuries, to a reversal of alliances, with France and England in the Entente Cordial, and the Ottoman Empire moving into German orbit. (England had traditionally allied with Prussia, and France with the Ottoman Empire.)

When World War I broke out in 1914, and the Ottoman Empire allied with Germany, France and Britain together moved to quash Ottoman influence in Egypt because of the strategic value of the Suez Canal. Britain was selected to be Egypt's "protector" because of her superior Navy, and her ability to project naval power into the Eastern Mediterranean. This completed the earlier Franco-British moves to weaken Turkish influence in Egypt beginning with the Greek war of independence in the 1820s.

In 1882, Ahmed Urabi, an Egyptian general, led a rebellion against the Egyptian Khedive, a viceroy to the Ottoman Empire, as at the time Egypt was an Ottoman vassal. The British had strong interests in Egypt, due to among many other things, the Suez Canal, and so, supported the Khedive. At about this time, the Khedive asked the Sultan of the Ottoman Empire for help, but received none. Then on June 11th, 1882, riots began, in which 50 European business men, and 250 Egyptians were killed. Though the actual cause of the riots were unknown, the British, blamed it on Ahmed Urabi, bombarded Alexandria, and landed a force in the Canal Zone, a area including the Suez Canal, and proceeded to wipe out Ahmed Urabi's army. On September 13, 1882 the British under Garnet Wolseley, fought and won the Battle of Tel el-Kebir. Ending Urabi rebellion, and restoring the Khedive to the throne. Though it was at first was meant to be a short term occupation, but was dragged out for many reasons, until, Egypt found it's self a protectorate, and later, a Dominion.

Egypt falling into England's grasp began earlier, with Napoleon.

By the end of the 18th century, the Ottomans, focused on fighting the Russians had focused their resources on protecting their European territories and were only weakly holding on to Egpyt.

Napoleon realized that even though the Suez Canal had not been built, the British were still able to land resources from India in the Red Sea and Persian Gulf, haul them across the Levant, and sail them to Europe on the Mediterranean. Napoleon directed the French army to invade Egypt and Palestine to deprive England of this ability.

After Napoleon was defeated, the Ottomans were not able to properly reassert themselves over Egpyt. Muhammed Ali, an Albanian mercenary general, established the Khedivate of Egypt which was an independent state in all but name. The British supported Muhammed Ali to prevent the Ottomans from ever reconquering Egypt. Muhammed Ali's British backed dynasty ruled Egypt until the overthrow of king Farouk in 1952.

Funny story, that. It all starts with the Suez Canal.

Shipping things between the far east and Europe the long way around Africa was certainly doable, but very very time-consuming and expensive.

Once built, the canal was half owned by the French and half owned by Egypt. However, Egypt's finances were your typical third world despotic mess, so in 1875 the ruler of Egypt was forced to put his one money-making asset up for sale: Egypt's half stake in the Suez Canal. Since it had become a vital link between England and its Indian colonies, England snapped it up.

Obviously this did nothing to stop the economic rot, and Egypt soon found itself so in hock to British and French banks, that the whole country was essentially put into their receivership in October of 1876. Of course the locals weren't very happy with this, so the inevitable revolt against the foreign creditors occurred in 1881. This threatened both the British banker's investments, and the now-vital canal link to India. So the British Imperial impulse immediately kicked into gear, resulting in the 1882 Anglo-Egyptian War, the seizure of both the canal and the country by British troops, and finally the country's absorption into the British Empire.