Tex Avery was, in an odd way, a living contradiction of the idea that funny cartoons are all about funny drawings. Avery's cartoons rarely had the most advanced animation or the most expressive drawings and poses. Except Preston Blair's contributions, I rarely find the animation in his cartoons to be all that distinctive (Michael Lah, one of his animators in the later years at MGM, told Leonard Maltin that they found that too many drawings distracted from the comedy and that they found it was better to keep the animation closer to the "pose reels," or what we'd now call an animatic). Even his famous "wild takes" can be a little drab as pure drawings. He wasn't bad at funny drawings, but that is not what makes him great.
What makes Tex Avery one of the greatest figures in animated comedy is his incredible expansion of the range of gags available to cartoons, what we might call the vocabulary of gags. He was not the first to take advantage of the potential of animation for re-shaping characters' bodies, defying the laws of physics, or breaking the fourth wall, but he made that the primary basis of an animated cartoon. Before Avery, a cartoon was basically a comedy short with gags inspired by silent comedy or comic strips, plus some gags that took off into an alternate cartoon reality. Avery, particularly after arriving at MGM, switched it around: instead of that kind of gag being a special effect, it became the kind of thing that happened all the time. At a time when Disney was pushing cartoons closer and closer toward a heightened version of real life, Avery found an alternate path: why not make cartoons where everything that happens could not possibly happen in real life?
That's why the most common type of Avery gag is one that takes a real-world object and changes its physical properties in some way. Characters appear in two places at once, cannons act like giant penises (Blitz Wolf), logs get peeled like bananas (Three Little Pups), a little duckling can pick up a boat with two hunters in it (Lucky Ducky). The more removed a gag was from our physical reality, the better Avery liked it. One gag he reportedly hated and considered hacky was the standard gag of a character crashing through a wall and leaving an outline of himself, and that happens to be a gag that is almost too close to what could happen in real life (I mean, it doesn't happen, but if the wall was thin enough... anyway we've seen a live-action character do that, Daniel Craig in Casino Royale). For Bob Clampett or Chuck Jones, in their different ways, animation was about funny drawings. For Tex Avery, it's more that animation was about funny things that are drawn, or, more specifically, funny things that can only be drawn. Even his wild takes are essentially jokes about body parts flying apart or changing shape or size. For Avery, it was always about how to do what can't be done in real life.
The insight Tex had was that the limitations of a short cartoon can be liberating. The limitations are clear: it's only seven minutes so there's no time for any real depth or complexity of storytelling, a short funny-animal cartoon is incapable of giving a direct, realistic representation of life, and it's the light entertainment before the feature so the audience isn't really all that involved. Avery turned these disadvantages to his advantage. Audiences didn't -- couldn't -- take a short cartoon as seriously as a feature, so Avery talked directly to the audiences, made light of the insignificance of short cartoons and (almost accidentally) got the audience more involved in the cartoon just by enjoying the little asides he threw in for them. And because short comedy cartoons are inherently unreal, he was able to invent gags that would never work in live-action, even the more surreal era of silent comedy that he and other Termite Terrace directors were influenced by. In live-action, if you try to defy the laws of physics with a gag, the audience may like it but they may also be distracted by trying to figure out how it was done. In animation, everything is fake, there are no real people, so no one will ask any questions if a character instantly comes back unharmed after his body falls apart.
Other animation people may have done more for the art of drawing but nobody did more to help create the rules of what we now think of as cartoon physics and cartoon violence. Before Avery, cartoon violence was basically Popeye and Bluto hitting each other a lot, or violence influenced by the Keystone Kops. After Avery, and the WB and Tom and Jerry cartoons that were influenced by his ways of using violence, cartoon violence had its own internal logic and became at once more gruesome and more fun to watch (because Avery's violence was so removed from the real-world effects of violence, even when characters did something realistic like shooting themselves). He defied the laws of nature and science and created his own rules for the universe. And we're happy to live in his universe.