The first part of this answer is a partly modified version of this post on pre-war fleet doctrine.

To understand how carriers gained primacy over the course of WWII, an understanding of how navies thought about carriers before the war will be helpful. All three of the world's carrier navies, the Royal Navy, United States Navy and Imperial Japanese Navy, used their carriers in very different ways. There was one constant, though. All three planned to use their carriers in subsidiary roles, supporting battleships in the battlefleet.

The RN had experimented with carrier-centric doctrines during exercises in the late 1920s and early 1930s. From 1929, the Mediterranean Fleet had exercised with the three carriers, Courageous, Glorious and Furious, acting together. These exercises clearly showed that carriers were more effective working together. While the RN planned to operate carriers together, it never fully developed the combined strike doctrine that the IJN did. Instead, the air groups of these carriers would operate more autonomously. The exercises of the 1920s-30s clearly showed the capability of the carrier. Equally, they were seen as being limited by the performance of available aircraft, and by the numbers of available carriers. They were also seen as being vulnerable in the event of a surface action, especially at night or in bad weather. To this end, carriers were seen as an essential part of a combined-arms tactical system. The RN's battle-fleet doctrine of 1939, as laid down in the 1939 'Fighting Instructions', assigned the carrier four main roles: finding the enemy fleet, sinking the enemy's carriers, to slow and damage the enemy battlefleet, and to observe the fall of shot for the battleships. Carrier aircraft were also seen as significant for anti-submarine work. The carriers were to be concentrated in a single force, along with the rest of the battlefleet. The carriers would protect the battleships from air attack, while the battleships protected the carriers from surface attack. The carriers would also benefit from the additional anti-aircraft umbrella provided by the battleships. However, the need for the whole force to manoeuvre as one slowed down the tempo of carrier operations. Once the enemy carrier force had been defeated, and their battlefleet slowed by carrier attacks, the British battlefleet would seek to fight a surface action. Once it had closed with the enemy, the carriers would drop out of the combined force. The battleships were to close to ~15,000 yards, and then engage by divisions, rather than in line as it had during WWI. The RN wanted to fight at night, at relatively short ranges, and to this end, the Fleet Air Arm (uniquely) developed a night-strike capability.

The Japanese doctrine, like the British, saw the primary role of the carrier as destroying the enemy's carriers, and then harassing his battlefleet, as part of a combined-arms fleet. However, they saw the operations of this combined-arms fleet somewhat differently. During the 1930s, the IJN mostly expected their carrier force to operate in widely separated groups, aiming to surround the enemy fleet. However, as time moved on, they realised that this made the carriers vulnerable to air attack, and concentrated them into a single group, starting in 1940. This was to operate ahead of the main battlefleet, rather than with it as the British equivalent would. This allowed the carriers to manoeuvre as needed to launch aircraft, without having to drag the rest of the battlefleet along with them. The ideal Japanese battle would start with the Japanese submarine screen spotting the enemy fleet. The enemy fleet would then be worn down by submarine, air and night torpedo attacks. Once contact with the enemy battlefleet was made, the Japanese battleships would fight a long-range gunnery duel (ranges of 35,000 yards or more), before closing to a 'decisive range' of 19-22,000 yards to finish the enemy off.

American carrier doctrine strongly resembled the pre-1940 Japanese doctrine. The carriers, rather than operating with the battle-fleet, operated semi-autonomously. They formed, along with heavy cruisers and destroyers, as a scouting force. Carriers did not generally operate together, but operated independently, spread widely over a large area. Once again, the role of the carriers was to scout, to win air superiority by sinking the opposing carriers, and to harass the enemy battlefleet. Exercises carried out in the 1930s were seen as showing that combined-arms attacks were the right way to use carriers. They suggested that carrier attacks at the start of a battlefleet engagement would both greatly increase the effect of the carrier attacks and the battlefleet's own fire, as the enemy could not easily respond to both. The American battlefleet was intended to fight at 30,000 yards or more, operating in line astern. It also developed special tactics to counter the Japanese (and British) battlecruisers, which could use their superior speed to 'cross the T' of the American line. It was only as the Essex class began to roll off the slipways, the USN became more able to switch to a carrier-centred doctrine. Once they could gather enough carriers to operate a massed 'alpha strike', the battleship was no longer needed as the primary surface combatant.

It should be clear from the above that the three main navies saw the carrier fitting a similar, subsidiary role. The main difference was that the British doctrine saw the carrier operating as part of a combined-arms force, while the Americans and Japanese saw it as part of a more autonomous unit. But why did they hold to this doctrine? Sumida argues cogently that the carrier was, before 1940-41, the carrier was not, on its own, a decisive weapon. To see why, we can look at the strike aircraft they carried. The RN was, in 1939, operating the biplane Swordfish torpedo bomber, and the underpowered Skua dive-bomber. The IJN had the D1A Susie for dive-bombing (with the Val just about to enter service), though the B5N Kate was an excellent torpedo bomber. The USN was operating the biplane SBC Helldiver, and the TBD Devastator. With the notable exception of the Kate, all of these aircraft were slow, vulnerable, and had comparably limited ranges and payloads. Until the carrier had more effective aircraft, it could not be a decisive tool. Additionally, relatively few of these aircraft could be deployed - until carriers like Hornet or Zuikaku entered service in 1940-41, carrier fleets were comparatively small. As such, the belief that the carrier could only play a subsidiary role was justified. It was not that admirals did not understand the carrier. Rather, the carrier could not meet the expectations that its later effectiveness would place upon it.

It's sometimes possible to overstate the effect carriers had on the naval war. While aviation had a significant effect on the war, much of this was land-based aviation. The first battleship to be sunk at sea by land-based aviation was Prince of Wales in December 1941. Carrier-based aircraft, despite their success at sinking battleships in port, wouldn't sink a battleship at sea until Musashi in October 1944. Over the course of the war, just two battleships (Yamato and Musashi) would be sunk at sea by carrier-based aviation. Compare this to the six sunk in surface engagements (Hood, Bismarck, Scharnhorst, Krishima, Fuso and Yamashiro) or the four sunk at sea by land-based aviation (Prince of Wales, Repulse, Hiei and Roma). Only when we add in ships sunk in port - six for carriers (Caio Dulio, Oklahoma, Arizona, Ise, Hyuga and Haruna) and two for land-based aircraft (Gneisenau and Tirpitz) - does the power of air power at sea become clear. For much of the war, carriers would operate as envisaged by pre-war doctrine: winning air superiority for the battle-fleet to operate in, and harassing enemy operations. However, the carrier would still decide many battles.

The British, who did not face carriers in their primary theatres - the Atlantic and Mediterranean - followed their pre-war doctrine almost to the letter until the formation of the British Pacific Fleet in 1944. It proved effective at both Matapan and against Bismarck. In the former battle, strikes from Formidable crippled the Italian battleship Vittorio Veneto and heavy cruiser Pola. While the battleship would withdraw, an Italian cruiser squadron would slow to cover Pola's retreat. This allowed the British battle-fleet to close and destroy them in a night action. The sinking of Bismarck is well known. A torpedo hit from Ark Royal's Swordfish jammed her rudders, allowing King George V and Rodney to close and sink her in a surface action. The carrier also found significant use protecting the battle-fleet from attack by land-based aviation, especially in the Mediterranean. However, once the RN had enough carriers, and once it was fighting in earnest in the Pacific, they began to adopt the more carrier-centric doctrine that originated in the USN.