Many nouns can be both countable and uncountable, depending on context.
An uncountable noun refers to an idea or a concept. A countable noun refers to a specific thing. So for example, I might say, "I like chocolate." I like chocolate in general. It's not that I like one particular piece of chocolate, but that I like chocolate in general. But if you were handing out candy and offered several varieties, say chocolate, caramel, and fruit bar, I might say, "I'd like a chocolate." I want one of the specific candies that is chocolate.
Same thing here. It would have been equally grammatically correct for the psychologist to have said "... develops into social anxiety." That is, "social anxiety" is a general concept. Shyness, which apparently she does not think is necessarily an example of social anxiety, could become one. But it wouldn't become just any kind of social anxiety. It wouldn't become fear of open spaces or fear of cats. It would become one particular kind of social anxiety, namely, an extreme form of shyness. It doesn't become social anxiety in general, it becomes one particular kind or example of social anxiety. That is, it becomes "a" social anxiety.
If the paragraph had not used an article, if it had said "... develops into social anxiety", a reader might possibly interpret that to mean that shyness can become all sorts of social anxiety, or can become social anxiety in general. I think most readers would guess that that was not what was meant. But that wouldn't be an impossible thing for someone to say.
The difference can be subtle, I freely admit.