By platonism, I will mean the contemporary view on it in analytic philosophy, and not necessarily Plato’s view.
Platonists believe that the existence abstract objects are needed to make sense of three phenomena: resemblance, subject-predicate discourse, and abstract reference. By contrast, nominalists think that we can explain the three phenomena without the extra ontology. Some candidates for abstract objects are properties, kinds, relations, propositions, numbers, sets, possible worlds, and states of affairs. (A platonist needn’t accept all these candidates into their ontology.)
Here is a short summary of those three phenomena. A more detailed treatment can be seen in Michael Loux’s Metaphysics A Contemporary Introduction.
Apples and roses resemble each other in their redness. Platonists think this resemblance is explained by the particular apples and roses both exemplifying the abstract object of redness.
Consider the proposition “Socrates is wise.” Platonists argue that in order for this proposition to be true ‘Socrates’—the linguistic object—must be correlated to a non-linguistic object—the person Socrates. Likewise, ‘wise’ must be correlated to a non-linguistic object. While the subject ‘Socrates’ operates as a noun—a singular term—the predicate ‘wise’ is an adjective and operates as a general term. That is, the predicate ‘wise’ can be true of many things like Socrates and Plato.
Corresponding to the adjective ‘wise’ there is the noun ‘wisdom’. Since objects are referred to by nouns, it seems like the use of a noun implies the existence of a non-linguistic object to be referred to. Nouns also play the role of the subject in a sentence. In its use as a subject in the sentence “Wisdom is a virtue”, it seems ‘wisdom’ is explicitly referring to an abstract object.
Loux, Michael J. Metaphysics: A contemporary introduction. Vol. 2002. London: Routledge, 1998.