Learn Finnish: Minä en tule, siksi että olen sairas. - I am not coming, because I am sick.

Miksi is the question word for "why". Siksi että is a common way to answer it, meaning "because" (literally "for that reason that"). If someone asks "Miksi sinä et tule?" (Why are you not coming?), you can answer "Siksi että olen sairas."

Could I use koska instead of siksi että here?

Yes! Koska is another very common word for "because". You could say "Minä en tule, koska olen sairas" and it means exactly the same thing. Siksi että just puts a little more emphasis on the fact that this is the specific reason.

Why is the verb tule instead of tulla or tulen?

This is the negative form of the verb tulla (to come). To make a negative sentence, you take the negative verb (en, et, ei, etc.) and add the verb stem. Since "I come" is minä tulen, you drop the -n to get the stem tule. So, "I do not come" is en tule.

Do I need the word minä at the beginning, or could I leave it out?

You can leave it out! Because the negative verb en already tells us who is doing the action ("I don't"), just saying "En tule..." is perfectly clear and very common in everyday Finnish.