The phrase is used to show how uneducated the speaker is. Since the sentence is indeed a double negative, it reflects on the speakers actual need for education, because they have bad grammar. It's a bit a of a play on words/joke.

Because it sounds good. Pink Floyd sacrificed good grammar for great music.

Well, your sentence is a double negative sentence (a sentence having two negatives). When a sentence is a double negative sentence, you can think it of as an affirmative sentence because two negatives cancel each-other (As in Maths, two negatives cancel each other). And the sentence "We don't need no education" would mean "We need education (DON'T and NO cancel each other)". I think you've got this! If not, feel free to ask.

The "double negative" is right on the boundary of "good English" and "bad English." An ordinary person of ordinary education, in ordinary colloquial speech, might well use the correct pattern, "I don't need any X." He might also use the incorrect double negative pattern, "I don't need no X." The double negative could be used for any of several reasons: a) The person really doesn't know better--didn't pay attention in school, doesn't need to or isn't able to speak correct English. b) The person feels so negative that any non-negative word feels wrong; the double negative is used for emphasis. (Like the Cowardly Lion in the movie "The WIzard of Oz: "You wouldn't be afraid of anything?" "Not nobody, not nohow!"). No, no! c) Under strong emotion, people forget what they've learned in school and revert to what they heard as children. d) The person WANTS to sound rough, because they are hostile to the world of "educated" people. You will often hear a pseudo-logical explanation of why the double negative is wrong. It is argued that in English grammar, "We don't need no education" OUGHT to mean "We don't need no-education" i.e. "We need education." The problem is that nobody hears it that way. Anyone hearing "We don't need no education" hears it as being incorrect English that means EXACTLY the same thing as "We don't need any education" or "We don't need education" or "We have all the education we need."

It simply fits the song. No need to read into it. It's funny how people focus on and "explain" the first line in the verse but ignore the second and fourth/fifth lines.