I can't understand this:
How do you catch a tame rabbit?
The tame way
I found that tame means:
not dangerous or frightened of people; domesticated.
But how does this fit into the joke?
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Half the joke is missing:
Q: How do you catch a wild rabbit?
A: Dress in orange and make a noise like a carrot.
It is "funny" because carrots don't make noise, and it results in a goofy mental image.
Now follow up with the joke in the question:
Q: How do you catch a tame rabbit?
A: The tame way.
It is "funny" because the listener is expecting an answer like the answer to the first joke — some sort of goofy visual imagery. Instead, they are given a little bit of wordplay, as described in other answers. The resulting cognitive dissonance provides hours of entertainment. (The unexpected answer is funny.)
I suspect the joke assumes knowledge of the Looney Tunes character Tweety Bird, and maybe Bugs Bunny. Tweety pronounces 's' sounds as 't', so if you asked Tweety how to catch a tame rabbit (vs catching Bugs Bunny, maybe), he would say 'The same way,' and it would sound like 'The tame way.'
Or maybe I'm reaching too hard. As Nathan Tuggy said, not all that funny.
Q. How do you catch a rabbit?
A. You sneak up on it
Q. How do you catch a unique rabbit?
A. Unique up on it
Q. How do you catch a tame rabbit?
A. The tame way, unique up on it.
I tend to regard the joke as a form of anti-humor. There's a little bit of word play going on, using homonyms to create something that is nonsense but sounds like something meaningful, and therein lies the amusement. It's along the same lines as a more famous type of anti-joke:
Q. Why did the chicken cross the road?
A. To get to the other side
Q. Why did the chicken cross the playground?
A. To get to the other slide
Q. Why did the chicken cross the ocean?
A. To get to the other tide
And there are many more of these kinds of jokes. Sometimes they take on a cultural value where people entertain each other by taking turns reciting them. They were all quite popular at my preschool.