Learn Farsi: من اینکه این شجره‌نامه تقلبی است را درک نمی‌کنم. - I do not understand the fact that this family tree is fake.

Because rā is marking the entire nominal clause as the direct object of the main sentence. The main verb dark ne-mi-konam (I do not understand) acts on the whole idea of the fact that this family tree is fake. In Farsi, you package that whole clause with in-ke, and then place rā at the very end of it to show it is the object.

Why do we have the word in twice in a row here (in-ke in)?

It looks like a stutter, but they serve different purposes. The first is part of the fixed phrase in-ke (literally this that, meaning the fact that). The second in is a simple demonstrative adjective meaning this, pointing to shajare-nāme (this family tree).

Could I put the main verb earlier and say man dark ne-mi-konam in-ke... like we do in English?

You will actually hear native speakers do exactly that in casual, spoken Farsi! Breaking the strict Subject-Object-Verb rule by putting long in-ke clauses at the end makes it easier to say on the fly. However, this sentence uses standard formal word order, keeping the main verb at the very end, which forces the big, heavy object clause into the middle.

What are the parts that make up the word shajare-nāme?

It is a compound word. shajare comes from an Arabic root for tree, and nāme means letter, book, or document. So a shajare-nāme is literally a tree document. You have likely seen used this way in other vocabulary, like (newspaper) or (The Book of Kings).