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'to come into its own' means that something has matured and found a useful place in the world. Ideas come into there own and people do to. All kinds of thing can come into their own, including, I suppose, poetry. I have no idea of the context in which this sentence was uttered but it sounds a bit ridiculous to me. Poetry is still a valid and popular form of writing but its heyday, or its 'coming into its own', has long since past. And really the sort of short dense phrases we see packed into text messages or internet memes hardly counts as poetry, or even a revival of the form, which is just my opinion.

The writer is saying that poetry has become significant and necessary in the information age. Poetry condenses ideas into a few words, and so it can convey a message briefly but powerfully. In the modern world, we are short of time and need our messages to be dense, concise, and to the point - this is why we should now be taking notice of poetry. In the past, when there wasn't this need for brevity, poetry was less important; in the information age, we need poetry more than ever before.