Translation:This disease causes him to have a seizure.
I understand the sentence now after learning that the phrase "neden olur" essentially means "leads to" or "results in".
So the breakdown of the sentence is as follows:
"Bu hastalık.... neden olur" = "This illness will lead to/will result in [...]"
"nöbet geçirmek" = "to have a seizure"
"onun nöbet geçirmesi" = "his/her having a seizure"
Therefore, you bring it all together and it becomes:
"Bu hastalık onun nöbet geçirmesine neden olur" = (literally) "This illness, (to) his/her having a seizure will lead/will result", i.e. This illness will result in him having a seizure.
This is how I understand it. 'onun nöbet geçirmesine' - nöbet geçirmek together simply means having a seizure but to understand it grammatically you have consider what the words mean separately. Geçirmek means to pass or undergo something. So it's onun (her) undergoing of a seizure = onun nöbet geçirmesi. The -ne suffix is because neden olmak needs a dative case ending on whatever it's acting on. The -n- is a buffer between vowels. Does that make sense?
Why 'neden olur' is translated into Future Tense? Since Aorist is supposed to be timeless, can it also be Present Tense?
Of course if 'seizure' stands for heart attack future makes more sense. But could 'nöbet geçirmesine' also mean some longer attack like an epileptic attack or a fever cramp or other convulsions? In this case you might talk while the poor guy is suffering and make a Present Tense statement. Or if it was some repetitive attack every two hours. Maybe then you also might use Present Tense.