Learn Spanish (Latin America): Hasta los animales del circo están cansados.

Yes, it frequently means "until" (as in hasta luego or hasta mañana). But in this lesson, you are seeing it used as an adverb meaning "even". It is used to express surprise or to point out an extreme case, highlighting here that even the animals are exhausted.

Could I use incluso instead of hasta in this sentence?

Absolutely. Incluso los animales del circo están cansados means exactly the same thing. In spoken Latin American Spanish, hasta is extremely common when you want to add this kind of dramatic emphasis.

Why is it están cansados and not son cansados?

We use estar because being tired is a temporary physical state. If you used ser, it changes the meaning completely: son cansados would mean the animals are inherently tiring or annoying to deal with, rather than feeling tired themselves!

Why do we say del circo instead of just de circo?

Del is a mandatory contraction of de and el. Using del circo literally means "of the circus," referring to the animals belonging to a specific circus the speaker has in mind. If you said animales de circo, it would refer to "circus animals" as a general category, rather than specific animals at a specific show.