The introductory scene. After Oxide gives his speech to the world, we are introduced to the main eight characters preparing themselves for the challenge. Tiny fixes his kart... by smashing it with a hammer. Polar is reading a "How to Drive" book upside-down. Dingodile's motor explodes in his face leaving it black. And Crash... is sleeping.
The fact that the racers have the same face all the time. Crash, Tiny, Cortex, Polar... all smiling. Even when Aku-Aku or Uka-Uka are talking very seriously.
The potential of interrupting Aku-Aku and Uka-Uka when they are giving instructions and suggestions.
During the races, when hitting rival racers. Using the unavoidable sphere when everyone is further than your character and hearing the chorus of "ouch", "ow" and "damn" from the other racers is simply hilarious.
Being a racing game where random things happen, funny moments can occur any time, e.g. the player is having fun with its friends at the multiplayer mode.
In meta terms, an interview reveals that Naughty Dog specifically created Oxide so that the series would jump the shark or, in their words, "We tried to kill Crash". Because they believed no one would take an alien seriously.