Learn English – What does “here’s to someone/thing” mean

"Here's to ..." is a phrase used when making a toast. It means "Here is a toast to ...", at which point it is customary to raise your glass. The phrase has a life of its own, used by a speaker or writer expresses a situation that calls for a toast-like salute to something.

The "Here is" part can be omitted also. For instance, the wording of a toast may simply be something like, "To your health!"

The preposition or particle "to ~" in this situation means something like: "for the sake of the (continued) good status of ~" or "in recognition/appreciation/celebration/congratulation of ~" or "in hope of ~"

"Here is to you, Bob!" -> "Here is a toast in appreciation of you, Bob!"

"To your health!" -> "Let this symbolic drink be made for the sake of the continued good status of your health!"

"Here is to finishing off the semester in a positive way."

-> "Here is a wish in hope of finishing off the rest of the semester in a positive way."

-> "Here is a cheer in celebration of having finished the semester in a positive way."