James Paris Lee's detachable box magazine was pretty revolutionary for the time; every other rifle was using a blind magazine within the action, or the Mannlicher enbloc system. Originally designed to hold five rounds, the magazine (single stack) was changed to hold eight rounds when the Magazine Lee-Metford was accepted into service in 1888 by the British Army. This was later changed to a double-stack 10-round design in about 1893 (the exact year escapes me right now) at the suggestion of one Joseph Speed, a manager at the Royal Small Arms Factory in the town of Enfield Lock.
Changes in rifling transformed the Magazine Lee-Metford into the Magazine Lee-Enfield, otherwise the rifles were more or less identical.
The transition to a magazine-fed repeating bolt action stems from the British getting thoroughly whipped by the Boers in various engagements during the First Boer War, 1880-1881. The British were equipped with single-shot Martini-Henrys, the Boers mostly were using Mauser rifles that afforded them a higher rate of fire and the ability to shoot over longer distances, both huge advantages.
Enfields were adapted to accept loading via chargers (stripper clips, invented in 1888 by Paul Mauser) from 1902 and onward, when the Short Magazine Lee-Enfield was introduced. Older rifles were updated with a bolt-head that was notched to accept chargers, or with the addition of the charger bridge. I've got a 1901 Magazine Lee-Enfield that was updated in 1909 with the addition of the bridge.
The Haldane Reforms of 1906 changed an awful lot about the British Army, and marksmanship was emphasized very heavily, in no small part because forward thinkers in the Army realized that machine guns were going to become increasingly important on the battlefield, and given the relatively small number of automatic weapons in stores at the time and their expense/slow acquisition rate, the next best thing is to have soldiers that are highly-trained riflemen that can lay down a fast, accurate volume of fire.
The rifle standards of the day (15 rounds on-target at 300 yards in a minute, "Mad Minute") demanded either a lot of reloading or a large magazine capacity, which the SMLE had. Couple that with the fact that five rounds on a charger was the de facto standard of the day, and two chargers makes for a full magazine, very convenient logistically. Additionally, the cock-on-close design of the rifle's bolt lends itself well to rapid fire.
As to why nobody else was using such a large magazine (an old Swiss rifle actually had a capacity of 11, the 1886 Lebel could hold 8 in a tube magazine, one in the elevator and another in the chamber for ten, some Mannlichers could hold six), I can't offer up a wholly conclusive answer, but between the fact that nobody else was using detachable box magazines on service rifles, nobody else built aimed rapid fire into their infantry doctrine, and a huge chunk of the world had more or less standardized on the Model 98 Mauser and it's internal magazine meant that the SMLE was in a league of it's own concerning rate of fire and overall capacity.