I think SPavel has written a very good answer. However, I think there are two other factors that should be mentioned, and they won't fit well in comments, so I'm adding a supplemental answer.

First, it's worth noting that the US differed from many countries in that it didn't really have a strong commitment to an existing weapon. It entered WWI quite late, and when it did, it was with a mix of M1903 Springfields and M1917 Enfields. The US also had only relatively recently switched from a previous service rifle (the M1892-99 Krags). Conversely, the other WWI combatants who became major WWII combatants generally had more resources invested in their bolt-action rifles (to the extent of my knowledge, of the major WWII combatants, only the US and Japan had adopted their WWI service rifles after 1900; most countries also predominantly used a single rifle as their WWI service rifle, unlike the US's split between the Springfield and Enfield). In other words, the US had less cost sunk into their existing service rifles than most countries, and thus less to lose by moving to a new design. (If standardizing on a single weapon means you already need to rearm half your army, it's not as large of a step to just rearm all of it; and if you're doing that, you might as well use a new design.)

The US also had considerable manufacturing capacity, making it easier to produce a large number of semi-auto rifles (which are comparatively more complex to produce and maintain than bolt-actions) in the time between the wars. (SPavel already noted the difficulty France had in producing the bolt-action MAS-36 in sufficient quantity, but it bears repeating as a point of comparison.) Coupled with the US's relatively late entry into the war (as SPavel also noted), they simply had the luxury of being able to rearm most of their military with the newer, more complicated design, while the required amount of production would have been difficult or impossible to achieve for other countries, especially when early-war combat losses were taken into account.

In addition to SPavel's excellent answer...

They were more expensive.

With its large manufacturing base, and at the time of adoption a very small army, the US could afford a more expensive rifle, but more importantly, it actually had one to adopt.

The US was starting from a relatively clean slate.

Other nations of WW2 had large standing armies with large stocks of bolt-action rifles left over from WW1. The Battle of France, for example, involved almost 7 million troops. Most WW2 armies used modified versions of their WW1 rifles. All the training, tactics, logistics, supply, maintenance, and manufacturing were in place. Moving to a new rifle would require replacing all that with a brand new system, while keeping the old one around during the transition.

In contrast, at the time the M1 was adopted (1936) the US Army in 1936 was anemic; less than 200,000 men and poorly equipped. M1 production was slow and there were many design flaws to be worked out, even if the development phase took over 16 years and had radical changes along the way (i.e. adoption of new .276 (7mm) cartridge, which was abandoned half-way). Curiously, the official requirement for semi-auto rifle was submitted in 1907 (IIRC), so it may be due to this long period that US alone had, actually, working semi-auto rifle for standard, full-size cartridge, in 1936 already.

When WW2 started the US military had to had to swell up to millions in a few short years, a transition that it already experienced in 1918 (when US army numbered 4.8 million). They did not have anything like enough service rifles in inventory for another such expansion. If you're going to make millions of rifles anyway, and you have the industrial power and money to do so, might as well make it the most modern you can get. In addition, the US Army took to heart lesson of Spanish-American War in 1898, where Spain forces were equipped with 1893 Mauser rifles. That's how M1903 rifle came to be in US Army (which can be called licensed K98), but due to public outcry in 1898, US Army adopted pro-active stance in arms procurement. That's why there were a number of programs aiming at delivering what we now call "next-generation" weapons systems, of which M1 Garand was just one.

Other armies relied on machine guns for squad firepower.

A German infantry squad was centered around a light machine gun, such as the MG 34 and later the MG 42. This made up for the relative lack of firepower of the Kar 98k bolt-action rifles most of the rest of the platoon carried.

The British had the excellent Bren gun for their infantry.

The US never developed a light machine gun for WW2. A soldier could lug around a Browning 30 cal (32 pounds) or a BAR with a 20 round magazine designed for the outdated idea of "walking fire". Both were of (slightly after) WW1 vintage, quite heavy, and inadequate for the light machine gun role.

Instead, by equipping everyone with an self-loading rifle the ordinary rifleman could pour out quite the weight of fire compared to their bolt-action counterparts.