Well, I think your question would be better phrased as "Why did the USMC get so large after 1941?"

It should be noted that the USMC is totally unique worldwide in terms of size. Every other marine corps in history has been quite small, and this is simply because of the traditional mission of marine corps.

Marines have traditionally acted as naval infantry, boarding/landing parties, and port security. Historically marines have not operated very far from coastal or littoral areas and have not operated on land for extended periods simply because of their role as naval auxiliaries. Because of this, there is simply no reason for any nation to ever have a large marine corps. In fact, since the USMC expanded, there has been a lot of controversy over why the USMC is as large as it is. It has been accused of acting as a second land army and a third air force (duplicating the US Army, USAF, and USN's capabilities), and for this exact reason no other marine corps has expanded like the USMC has.

Now, as for why the USMC expanded, a lot of it has to do with the US's need for mobile, rapid response forces post WW2. Naturally, the USMC's relationship with the Navy gave it a unique position to be that forward rapid response force. The US Army has traditionally been the branch with vast, heavy, mechanized forces (almost all of the United States's artillery and armor) that can take a long time to mobilize. Outside of a few units (like the Airborne or SF), the Army can take 30 days or longer to muster its forces for combat. The USMC, under the expeditionary force model, is supposed to be able to provide forward deployed forces that fill the gap in time until heavier Army units can arrive. Marine Expeditionary Forces are designed to sustain combat for about 30 days at a time, which is about the time heavy Army units will take to get to theater.

Now, whether or not this model is still relevant to today's realities is probably outside the scope of this subreddit. But in a nutshell, the USMC was always small because that's what marine corps are traditionally supposed to be according to their missions. The USMC only later expanded because of the need for rapid response forces, but the debate continues on shrinking the USMC because they duplicate many of the capabilities already provided by the other branches.

There are a number of facets to this question, so I'll try to cover as much of it as possible from my background knowledge.

In World War I, the USMC was used exclusively as traditional land based infantry. A marine really didn't carry out missions any different from an army soldier (read about Belleau Wood and other USMC engagements, and then compare them to engagements in which the Army was involved. Either branch could have just as easily fought the other branches battles), so after the war, the Army started to question whether the Corps deserved to exist on its own.

The Army argued that the USMC should be rolled into the Army, acting as a division or group of divisions with specialized roles within the army (Similar to how the 10th Mountain operates today, or the 82nd/101st Airborne: division sized units with specialized training and roles). The USMC, as you can imagine, didn't like this at all, and wanted to stay independent.

In between the wars, there were a series of War Plans that were color coded. Each War Plan was detailed a theoretical war, and how the US might go about prosecuting said theoretical war. These involved wars with Germany, the UK, Canada, various Central and South American countries, Carribean countries, and, most importantly for this question, Japan.

Here's some more info or additional resources on these plans: link 1 link 2 link 3

War Plan Orange link 1 link 2 was the plan detailing a hypothetical war with Japan, and this was the War Plan that the USMC went all in on. I can't say how serious the ideas of folding the USMC into the Army were, since I've never seen any primary sources detailing these conversations, but War Plan Orange was the Corps' baby.

War Plan Orange more or less necessitated the existence of the Marine Corps, and not just as a small specialized branch of the Army, but as an entire multi-division structure in and of itself, because it projected the necessity of a series of assault landings on islands, and infantrymen who are specially trained and equipped to carry out such landings.

This idea, of opposed landings on defended beaches, would become the bread and butter of the Corps in between the wars. The Higgins Boat (LCVP) was even developed through a series of trials in the 1930s in order to facilitate these assault landings.

With the start of World War II, War Plan Orange more or less came true, and the USMC was significantly expanded pursuant of this goal of conducting large scale, and largely independent operations.

So, that answers why it got so large after World War I. Before that was a different question, and something that I admittedly know less about (and don't have sources on hand for unfortunately).

For starters, the historical role of the Marines (and really the Marine forces of any country), was as on-board infantry complements on ships to 1) ensure there were trained soldiers on board to conduct both offensive and defensive boarding operations (rather than giving guns and other weapons to poorly trained sailors), and 2) to forcibly prevent mutiny.

Neither of these tasks really require large complements of troops to accomplish. Additionally, the US Navy didn't really start to swell up in size until the fleet overhaul/expansion program spearheaded by Alfred T. Mahan in the second half of the 1800s. link link link link link.

The USMC didn't do much, and therefor didn't need to be big. The famous fighting in the Barbary War? There were a total of 54 Marines and soldiers in the theater. At the Battle of Derne, there were a grand total of 8 Marines on the ground, leading a force of mercenaries. This is because there weren't that many Marines available. The US committed ~15% of their total manpower available in the theater for the Battle of Derne, which was the aforementioned 8 people.

The USMC wasn't used very often, simply because they weren't really needed that much. You didn't conduct opposed assault landings in those days, you just went where the other guys weren't and offloaded your army.

Long story short, the USMC didn't really come into its own until War Plan Orange gave it a purpose. Before then, it was confined to either providing security for Navy ships, or doing the same exact thing that the Army did (fight conventionally on land). It wasn't until they created this defined and specialized purpose for themselves that the USMC became something that needed to be big.

Some additional sources:

Craven, Wesley F. and Cate, James L. eds. The Pacific: Guadalcanal to Saipan: August 1942 to July 1944, vol. 4 of The Army Air Force in World War II. Chicago: University of Chicago Press for the Air Force Historical Division, 1950.

Hunter, Kenneth E., and Margaret E. Tackley. The War against Japan. Washington, D.C.: Center of Military History, U.S. Army, 2006.

Miller, Donald L. D-Days in the Pacific. New York: Simon & Schuster Paperbacks, 2005.

Miller, Edward S. War Plan Orange. Annapolis, MD: Naval Institute Press, 1991.

Sandler, Stanley. World War II in the Pacific: An Encyclopedia. New York: Garland Pub, 2001

Strahan, Jerry E. Andrew Jackson Higgins and the Boats that Won World War II. Baton Rouge, LA: Louisiana State University Press, 1994.

Rottman, Gordon L. World War II Pacific Island Guide: A Geo-Military Study. London: Greenwood Press, 2002.