Learn Finnish: Oletko innostunut uudesta projektista? - Are you enthusiastic about the new project?

This is because of adjective rection. The adjective innostunut (enthusiastic or excited) requires the thing you are excited about to be in the elative case (-sta or -stä). In Finnish, you are enthusiastic 'out of' something!

innostunut looks like a verb form. Is it an adjective here?

Yes. It is technically the past active participle (often called the NUT participle) of the verb innostua (to get excited). However, it is used here exactly like a regular adjective meaning 'enthusiastic'.

Why does the word uusi change to uudesta instead of just adding the ending?

The word uusi (new) undergoes a stem change. For many older Finnish words ending in -si, the stem changes to -de- before adding endings. So uusi becomes the stem uude-, and adding the elative ending gives us uudesta.

Why are both uudesta and projektista in the same case?

In Finnish, modifying adjectives must agree in case and number with the noun they describe. Because the rection rule for innostunut forces the main noun into the elative case (projektista), the adjective describing it must also take the elative form (uudesta).