Learn Turkish: Bizim gibi, onların da projeyi desteklemesi önemli. - Just like us, it is important for them to support the project.

This is an action clause acting as the subject of the sentence. It literally translates to "their supporting." To build it, you make the subject a possessor (onların) and add the short verbal noun suffix (-me) plus a possessive ending (-si) to the verb stem. The entire chunk "onların projeyi desteklemesi" becomes the subject of the word önemli (important).

Since "onların" is plural (their), why is the suffix on the verb "-si" (his/her/its) instead of "-leri" (their)?

In Turkish, when a plural pronoun like onların is explicitly stated, adding the plural possessive suffix (-leri) to the noun is optional. It is very common and natural to just use the singular possessive suffix (-si) to avoid repetition. Both onların desteklemesi and onların desteklemeleri are grammatically correct, but the shorter version is often preferred.

Why is "proje" in the accusative case (projeyi)?

Even though desteklemesi is acting as a noun (their supporting) in the main sentence, it still behaves like a verb inside its own clause. Because you are supporting a specific thing, proje takes the specific direct object (accusative) marker -yi just like it would in a regular sentence.

Why does the sentence use "bizim gibi" instead of "biz gibi" for "like us"?