Mainly because of his dialogue the Symposium, which is effectively a prose drama showing several people making speeches about the nature of love/desire. All of the speakers are in agreement that physical desire is the "lowest" kind of love, and as the drama progresses there's a kind of escalation of what "higher" kinds of love look like. One speaker casts "higher" love as a relationship between minds, not bodies; another casts love as a cosmic force, representing all forms of attraction; another casts love as the desire for wholeness; Socrates (the hero) casts different kinds of love on an ascending scale that reaches up, at its most philosophical "heights", to the meditation on and contemplation of abstractions that are usually known as Platonic forms.

The term "Platonic love" was coined by Ben Jonson in his 1631 play The New Inn. Act III scene 2 is essentially a discourse on love, and explicitly summarizes a number of passages in Plato's Symposium. At lines 73-6 the character Lovel starts off by defining love as follows:

For, what else
is Love, but the most noble, pure affection
of what is truly beautiful, and faire?
Desire of union with the thing beloved?

This is essentially a paraphrase of an opening section in Socrates' speech. A few lines after that, Beaufort paraphrases Aristophanes' speech in the Symposium, and Lovel makes the attribution explicit ("It is a fable of Plato's, in his Banquet, / and utter'd, there, by Aristophanes"). Lovel goes on to cite ideas from other speeches in the Symposium:

Beaufort. I relish not these philosophicall feasts;
give me a banquet o' sense, like that of Ovid...

Lovel. They are the earthly, lower forme of lovers,
are only taken with what strikes the senses!

And so on, until we get to Lovel coining the phrase at line 278, with the twist of simultaneously demanding a kiss as a fee for his time:

Lovel. Most Socratick Lady!
Or, if you will Ironick! gi'you joy
o'you Platonick love here, Mr Lovel.
But pay him his first kisse, yet, i'the Court,
which is a debt, and due: for the houre's run.

Plato's/Jonson's philosophization of love, and Jonson's emphasis on the non-sexual nature of "Platonic" love (a view that Plato did not share!), provided a useful way for people to rationalize homosocial relationships and categorize them as non-sexual. A lot of western homosocial culture depends on the existence of that non-sexual category. Platonic love as sexual did get somewhat rehabilitated, largely thanks to the invention of homosexuality and elevation of "Greek love" in the 19th century, but Jonson's phrase has stayed with us.