Maybe it's idiomatic and it doesn't have an explanation. I don't know, but perhaps someone else knows! :)
It might come from an expression like "μετά το τέλος αυτού του χρόνου" (after the end of this year) and shortening it to just "του χρόνου" to mean 'next year', but that's only a guess.
Ah - think I've figured it out (at least partly). I've noticed that the uses of the dative case in Ancient Greek generally get given to the genitive case in Modern Greek (e.g. σου δίνω το κρασί - I give you the wine / I give the wine to you), and I see from Wikipedia that this is indeed the case. So for example this explains why the toast και του χρόνου means "to the future". Dative in temporal expressions in Ancient Greek means time when, so I guess του χρόνου is "in the future", but of course also "in the (next) year". So that's my guess, but I'd love to know the actual answer.