What were Plato's worries?
Plato certainly worried about governments censoring intellectuals. This he discusses in The Apology (or just Apology) in which he defends Socrates who had been charged with corrupting the youth of Athens.
Plato was concerned with the general state of governments and citizens to the extent that he wrote The Republic in which he encouraged all to seek the Absolute Truth (notably in "The Allegory of the Cave") and in which he encouraged politicians to become more philosophical. He even encouraged a state in which "philosopher-kings" would be among those in charge.
Although Plato used allegories in his dialogues, he favored dialectics (logic) over poetry. In fact, Plato, usually via the character of Socrates in Plato's writings, criticized poetry. Plato believed that the absolute truth of all things lies in a realm of Ideal Forms. These Forms are abstract, unchangeable, and perfect in their immutability. For example, there is the abstract, ideal chair. Then there is the physical chair that you sit on. Then there is the chair painted by a painter or a chair described in a poem. The Ideal Chair is the perfect chair. Each reincarnation (physical chair, chair in a poem) is one step more removed from the true, ideal chair. Based on this thinking, Plato was skeptical about poetry and even writing itself. This is ironic because Plato used writing (obviously) himself.
And although Plato was not completely against writing, he had a logical skepticism about it (then being like a new technology) as we do today about technology. With the advent of new technology in the modern era, we don't have to remember phone numbers, bills are paid automatically, and if we don't know something, we don't read books, we can just google it. Plato had similar reservations about poetry and writing. If things are written down, we don't have to remember them.
Plato's criticism of writing (and poetry) is shocking to modern readers unless you consider the analogy with technology today. Writing, for Plato, and computers for the modern citizen, both offer convenient ways to communicate and archive data. But the downside is that the written and the digitized do the remembering/knowing for us. Plato also felt that writing was flawed because it had no author to support its claims. In Phaedrus, Socrates says:
The offsprings of painting stand there as if they are alive, but if anyone asks them anything, they remain most solemnly silent. The same is true of written words.
Not only is it like a silent author, but writing, like poetry and painting, is external to the mind; away from the source/author. Likewise, writing and painting are thrice removed from the Ideal Forms of ideas, further away from the absolute truth. Plato simply worried that too much use of writing would lead to loss of the ability to remember and too much focus on things so far removed from the truth of Ideal Forms.
(In the 20th century, Jacques Derrida, citing Plato, would show how writing can be seen as a supplement, not a detriment, to speech and thought.)