I once wrote a short paper on rhytons and similar for art history class. The general category is called "plastic" sculpture if anyone is interested. From what I recall they were often used to aerate wine, which was poured through and collected in a wide kylix or similar. This was likely representative of even older traditions of animal sacrifice with the wine substituting for blood, which is why a bull's head is such a common shape for a rhyton. Over time some of them became more complex and ornamental rather than functional, and those are often the most beautiful museum examples.

It was actually a bit difficult to find information at the university library at the time so I had to go to the specific art museum library. And with a quick glance online it appears some of the info out there may be wrong. e.g. Wikipedia says "Rhytons modeled after animals were designed to make it look like the animal was drinking when the vessel was being filled", which doesn't really make any sense for the shape.

https://en.wikipedia.org/wiki/Amphora

https://en.wikipedia.org/wiki/Rhyton

Bucephoron