Learn Finnish: Ne on juossu metsän reunaan asti. - They have run all the way to the edge of the forest.

The postposition asti (meaning "all the way to" or "until") dictates the case of the word before it. When indicating the endpoint of a movement or a time period, it requires the illative case. So, reuna (edge) gets the illative ending -an, forming reunaan asti.

Could we use mennessä instead of asti here since the lesson is about both?

No. While both words express limits, mennessä is strictly used for deadlines in time (meaning "by a certain time"). Because the edge of the forest is a physical location in space, you must use asti (or its synonym saakka).

The translation says "They have run". Why doesn't the Finnish sentence use He ovat juosseet?

This sentence is written in colloquial (spoken) Finnish. In everyday speech, the pronoun ne (which standard grammar reserves for animals and things) is almost always used for people instead of he. Additionally, spoken Finnish pairs plural pronouns with the singular third-person verb form, so ovat becomes on.

Why is the verb form juossu rather than juossut?