This is probably a better question for r/AskLiteraryStudies, but the basic issue Plato has is with the mimetic (and therefore somewhat "counterfeit") nature of writing itself.

If one's goal is to understand the "ideal form" of something, when attempting to do so through writing one is now thinking not of an imperfect manifestation of that thing in the phenomenal world, but instead a facsimile of that imperfect manifestation. Writing is (for Plato) a copy of a copy and therefore both highly suspect in its veracity, and (perhaps more importantly) easily given over to misinterpretation.

As an example, take beauty. Plato would argue that while one may never be able to fully understand true beauty as an ideal, surely it must be better to look at (and directly experience/think about) something beautiful, as opposed to reading or writing about the very same thing?

This is also why Plato argues that writing will cause "memory to atrophy"; we are contenting ourselves to learn through the imperfectly communicated experiences of others instead of striving to learn through reflection on our own experiences. Rather than preserving our thoughts as we might assume it does, writing (for Plato) too easily takes the place of the thinking we should be doing ourselves.

I highly recommend looking at the entries for Plato and Aristotle in Norton Anthology of Theory and Criticism, it will provide a great introduction to the "debate" they had over writing's value in the creation of a "just" society.

r/askphilosophy would be a good place to ask this question.

In addition to u/Jurka_Madiq's answer, which is good, I would say that determining the intentions of statements in Plato's works is notoriously difficult. Firstly, there is the issue that while the statement you quote is written by Plato, it is not in Plato's voice, so to speak. The section you quote is roughly 275a in Phaedrus and it is spoken by Socrates to Phaedrus but is is referencing a story where the the god Theuth is speaking to the king Thamus. It is legitimate to ask the extent to which this third hand (at least) saying is meant to be taken as Plato's own view. It reflects Theuth's view, and presumably Socrates, as he is the one saying it in the dialogue - but does Plato want us to agree with or simply consider this perspective.

There might be some legitimacy in attributing the sentiment at least to Socrates. Socrates never wrote anything down after all. Plato did though, so on one level at least he can't fully agree with the idea. Further, in other dialogues Plato's characters can talk about the benefits of writing (in Laws 891a The Athenian says that having things like laws written down can be useful for things like reference).

The entry on Plato in the Stanford Encyclopedia of Philosophy (a peer reviewed online encyclopedia) touches on these issues, especially in sections 4-7: https://plato.stanford.edu/entries/plato/. Alternatively, any introduction to Plato (e.g., Julia Annas' Plato: A Very Short Introduction, Oxford, 2003, is good) will also discuss these matters.

One thing that many of the introductions to Plato will say, and this links back to u/Jurka_Madiq who gave a very different answer to me but one that I can legimately claim to be good at the same time, is that there are often many levels of interpretation to Plato's works. Indeed, it is likely that he wants to get people to think about the issues and engage with them. So on one hand, it could be true that he doesn't think writing brings us to the same understanding of truth as active philosophising does, while it also being that case he does think it has value in engaging us with a debate that sets us on the path in that direction.