Humor's on a different level than other forms of writing when it comes to readability. In fiction you can afford to be elaborate, but humor needs to fire right out of the gate or it'll flop. That being said, it's all too easy to pass over unneccessary words that make a joke weaker.
How do you catch these? Simple. Just read it out loud and use your head. I've mentioned before that written and spoken humor flows differently, and what constitutes "unneccessary words" in either of these mediums is different. Spoken humor needs to get to the point while written humor can lollygag a bit. Still, if you can cut out any extra words, the better.
Read humor aloud. See what words you can trim. Remember, you need to convey all the information the audience needs to understand a joke in the smallest amount of space possible. Besides trimming words, can you replace bigger words with smaller ones? Can you pare down phrases to their bare essentials? You're not dumbing it down, per se - think of it as removing the joke's barrier to entry, so the most amount of people can enjoy it. That can be a fun challenge in its own right.
Like all writing, make sure you hammer it out first before revising. It can be tempting to capture the flow of the words as you go, especially for something as short as a joke. Resist the urge, get everything else done first, then go back and clean it up. You'll be glad you did.
Until next time, keep on laughing.
Do you have any questions about humor writing that you want me to answer? What would you like to see me cover in future columns? Leave me a comment telling me what you'd like to see!