Very simple: It made books accessible. Throughout the later middle ages, the cost of books was one of the biggest limiting factors on getting an education. It's a frequent trope in medieval literature to find a poor, starving university student, who spends every penny he can beg, borrow or steal on books, and even given that, those collections might number only a few works. A few dozen books would be a respectable collection for a wealthy scholar, and collections numbered in the hundreds would be the privilege of only large institutions and the extremely wealthy. People even in the educated classes might go their entire lives without seeing a thousand books, let alone in one place. Secondary markets for used textbooks sprung up in university towns, but even these were beyond the means of many would be students.
The radical change the printing press wrought cannot be overstated. I have in front of my a source on the book production of medieval Europe. Their numbers are specific enough to make my medievalist heart suspicious, but I think the overall trend holds. Their data shows that Medieval European Print production in the second half of the 15th century more than doubles the Manuscript (Hand Written) production for the entire century, and then increases sixfold more going into the first half of the 16th century. There's substantial margin for error on those numbers, but the simple fact is that the supply of books was at least an order of magnitude greater in 1550 than in 1450. This means, necessarily, that the cost of books plummeted. While they remained, at least in those earlier days, something of a luxury good, they became a luxury available to far more people.
The printing press radically accelerated the process of transition from a primary oral society that started in about the twelfth century in Western Europe, and it's import to understand how fundamental this change really is. Books and writing are a tremendously efficient and effective way of storing and transmitting information, and mass access to that kind of information allows for all sorts of things. Firstly, the massive expansion of the administrative state we see in the 14th and 15th centuries would have been impossible without record keeping, and a large body of trained, educated professionals capable of dealing with those sort of documents. Those states required new, more clear law codes, trained judges to run them, and so on and so forth. All this meant is that powerful people had a strong incentive to promote literacy and education, so as to expand their base of power.
Literacy also works changes on the lives of people who are not ruling nations, though. For one thing, literate and oral storytelling tend to be very different. Oral stories are generally rooted based in the cultural memory, myth and ideas of the folk, and the composition process tends to be based in reconfiguration, modification, and adaptation of those traditions by storytellers. Formula, rather than being derided, is an essential part of the process, both as a practical aid to memory, and as a way to summon up the meaning of other stories in something new. A particular way of referring to a mythical creature could serve as a way to evoke ideas about it's nature, and might be brought up whenever those traits are needed. Athena in Homer, for example, is often referred to as "Bright-Eyed Athena," perhaps alluding to her wisdom, or connecting her with her animal, the owl.
Because a book can be checked and compared, and returned to in a way that an oral performance cannot be, it reduces the need to retell old stories, and eventually the literary ideology can change in such a way that doing so is undesirable (Telling someone else's story in a slightly different way might land you in court these days, an idea that would be utterly bizarre to an oral folk performer). This leads to another idea, which is perhaps even harder to wrap one's mind around: Literacy can change our ideas about truth.
Truth, as Westerners commonly understand it, is rooted in correspondence. A statement is true if it corresponds to the natural world. The fundamental true statement is something like "The sky is blue, grass is green." To borrow a term, this is "Scientific" truth, and it shapes our modern idea of truth in a really fundamental way. We expect our truths, even less obvious ones, to be universal and center-less (The sky is blue to all people in all places), and much of Western Philosophy has been rooted in dealing with, and usually trying to argue for the idea of universal, objective truth. Fundamentally, literacy gives access to this sort of culture, because it allows facts and information to be reliably stored and required.
(Side note: Before I talk about how Oral Cultures can be different, I want to be clear that this should not be taken as a value judgement. Oral ideas about truth are not "worse" because they "lack the technology" of literacy. Instead, while literacy is probably a precondition for the scientific model of truth, there is not a necessary, teleological progression from one to the other. Indeed, the idea of a teleological improvement in both technology and society in unison is itself a western ideological predisposition, not an obvious historical truth.)
Truth in Oral Communities tends to be more rooted in the ideology of the community, the memory of its members, and the moral implications of that truth. The clearest way to understand this is the difference between Folk and Written Law, at least in Europe. In early medieval Europe, disputes were generally settled and law made by consultation with the powerful and influential men in the community, and consulting the memories of senior members of the community about how similar disputes had been handled in the past. Ownership of land would not come from a deed or title, but from the fact that a man (Almost always a man) and his ancestors had lived on given land for as long as anyone could remember. What this means is that the law shifted over time as memories and people changed, and the moral nature of a dispute was necessarily considered. The "truth" of the law was what the influential people in the community though was the best way. Written law, on the other hand, as we know well, takes only itself into account. Law is true because it corresponds to the written codes created by political authorities. Everyone is (supposedly) equal in the eyes of the law, and it (supposedly) treats everyone the same way. Justice is supposed to be blind.
If this is interesting, u/Snapshot52 has written a lot on this sort of difference from an indigenous perspective, both in his own answers and in Monday Methods threads, such as this one. He is both more educated and eloquent about this sort of thing than I am, so I highly recommend his work.
In short (and I am sorry, this became very rambly) the printing press was successful because it was an incredibly helpful tool in an ongoing process of transformation in the Middle Ages. Powerful people desired a more literate, more educated populace to help them expand their power, and those educated formed a class of their own that wanted to expand through the widening of education. This change, from oral to literate (which again, the press did not start, only helped along), resulted in a whole host of changes that I have only scratched the surface of here.
Buringh, Eltjo, and Jan Luiten Van Zanden. “Charting the ‘Rise of the West’: Manuscripts and Printed Books in Europe, a Long-Term Perspective from the Sixth through Eighteenth Centuries.” The Journal of Economic History, vol. 69, no. 2, 2009, pp. 409–45.
Green, Richard Firth. A Crisis of Truth: Literature and Law in Ricardian England. Univ. of Pennsylvania Press, 2002.
Hellinga, Lotte, and J. B. Trapp. The Cambridge History of the Book in Britain. Volume 3 Volume 3. 2014. Open WorldCat, http://0-dx.doi.org.pugwash.lib.warwick.ac.uk/10.1017/CHOL9780521573467.
Reynolds, Susan. Fiefs and Vassals: The Medieval Evidence Reinterpreted. Reprint, Clarendon Press, 2001.