Consider this sentence:
I understand that 的 permits creating adjectives, like 我的 would be "my". But here, I can’t see how it works. What is its purpose?
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A reasonable translation of "你说的" is: "That which you spoke". 的 turns 你说 (a verbal phrase) into 你说的 (a noun).
你说是什么 doesn't make any sense grammatically - it means "You said is what?" You can say, "你说什么" - which is literally, "You said what?" or in proper English, "What did you say?"
你说的是什么, however, means "That which you said is what?" or properly, "What is that which you said?"
In addition to @Kang Ming's excellent answer.
的 is not only used as a possessive suffix (as in
我的). There are other uses, don't be confused.
In case of
的 serves as selector by an attribute. Literally, someone asks, "what is it?", but out of all possible contexts picks up a context of the recent speech.
你想买什么样的手机？ What kind of cell phone you'd like to buy?
很贵的。 A very expensive one.
There's another usage of
的, with the meaning of "certainly". See this answer for details.
One other way to look at it: 我的 is not only "my," but can also be, "mine."
I'm full, so please eat mine.
Just like in English, in Chinese you could end the sentence with either "my <noun>" or "mine". When you hear this sentence and notice no noun or noun-phrase follows 的, this is perfectly OK, and you can translate the 我的 as "mine".
But unlike English, Chinese follows a pattern. You can put other things before the 的, too. So not only can you specify 你说的事情 "The issue you're talking about," you can instead say 你说的 "That which you're talking about," which is awkward in English but concise and natural in Chinese.
The role of “的” in this case is similar to "what" in English. So
你说 -> you say 你说的 -> what you say (or said, depending on context) 你做 -> you do 你做的 -> what you do (or did, depending on context) 你说的和你做的不一致。 -> What you said does not match what you did. 他说的确实在理。 -> What he said really makes sense. 情况比我们估计的更糟。 -> The situation is worse than what we estimated. 你说的是什么？ -> What is what you said?
Or more concisely:
你说什么？ -> What did you say?
Or even more concisely:
什么？ -> What?