"To" is a preposition here. It's actually called a 'stranded preposition' in your example. That's because prepositions normally precede a noun or some other element called a 'complement' (cf. "You should give it to that person"), but the complement of the preposition "to" is missing in your example, so the preposition is said to be 'stranded'.
If we were to analyse your sentence, we would have to show what the missing complement is. In this case the complement is "who", so this is what we get:
"Who (i) you should really give the gardening prize to _(i)_ is the person with the best window boxes".
The notation "i" (after "to") is cross-referenced to the pronoun "who" to show that it is the missing complement.