Platonic is a term with plenty of meanings. This essay will not be about platonic affection and conservatism; it's about platonism as philosophy. Specifically, Plato's theory of forms.
I won't go into detail about his beliefs here; that's what the link was for. The relevant fact is that Plato emphasized that concepts that humans believe in have an essential existence outside of how we perceive and define them, which he often called "Forms". So, for instance, you have a Form of the number two, and a collection of two objects is an instantiation of that Form.
What the heck does this have to do with modern political conservatism, I hear/imagine you asking? Good question.
How often do conservatives oppose same-sex marriage on grounds that it's a violation of the "sanctity of marriage", or even just that it "changes the definition of marriage"? Very often. And liberals like me often don't understand what is even meant by this. Marriage is a human social construct. If you want to define marriage, just look at the set of all marriages that exist. That's the definition. Legalizing SSM changes the definition, sure, but why does it matter?
Many conservatives don't see it that way. They have a platonic concept of marriage, an actually existent object of which real-life marriages are reflections. This is why they say things like "even if the law changes, gay couples' relationships still aren't marriages", which makes no sense from a non-platonic perspective.
[work in progress]