Per title. Undergraduate tuition often costs tens of thousands of dollars per year, but postgraduates are commonly paid a stipend. Not a lot of money, but still much better than having to pay for tuition too.
Why do undergraduates pay, but postgraduates are paid? A first guess is that postgraduates contribute to the "output" of the department so they deserve to be paid, but that doesn't seem like a complete explanation because the number of students taught is also an "output".
You're looking at it the wrong way. Both undergrads and grad students are charged tuition by the university. This tuition is used to pay faculty, administration, operating costs, etc. Some undergrads receive full or partial scholarships. You might say they are not "paying" for undergrad, but the university is still getting paid, just from a different source. The same occurs with grad students; many of them are given research assistantships, meaning that their advisor found external grant money and is using it to pay both tuition and a stipend to the grad student. Grad students without teaching or research assistantships, such as many MBA or law programs, still have to pay tuition out of their own pockets.
Teaching assistantships are not external money, but essentially a department is saying that they have to pay someone to provide instruction, so it's cheaper to pay grad students to assist a professor than to hire more faculty. The university is still getting the tuition paid by the department.
PhD students are paid to do work.
Undergraduate students pay money to learn.
There is nothing similar about these situations. You can tell because PhD students who do their research and teaching work, but do not learn anything (perhaps because they knew the material it already) continue to get paid. A PhD student who learns but does not do teaching or research work will not continue to be paid. A portion of the PhD students' pay may be in the form of education.
(US-specific) As was pointed out in the comments, grad-level professional programs generally do charge tuition, often a lot of tuition. Many academic master's programs do too, as do arts master's like MFA and MMus.
But very few people would make the commitment for an academic PhD if they had to go deep into debt to pay 5+ years of tuition in order to earn an assistant professor's salary (if they're lucky) at the end of it. Even lower-ranked law schools have been having trouble attracting students because most lawyers just don't make enough money to pay back the loans.
Universities can recoup some of the cost of having grad students by using them for teaching and lab duties, but having a grad program is overall a money-losing proposition. They do it because it enhances the reputation of the university and because faculty want to teach grad students (and want grad students to do the work they don't want to do).
In addition to the other fine answers here, I think a more general answer is that, as with other markets, price is determined by supply and demand (and any distortions in the market). Looking at things from this economic perspective, the main reason why postgraduates are more likely to obtain substantial amounts of scholarship support than undergraduates is that postgraduates have far better employment options than undergraduates, so you need more money to attract them away from these available alternatives. (In economic parlance, the supply curve for postgraduate 'labour' is higher than for undergraduates.)
