He/she wants me to go to cinema. Literally it means "He/she wants my going to cinema". "My going" can be used here. This i suffix makes the object accusative. The verb istemek wants object to be accusative here.

Now you can ask why we cant say "Sinemaya gitmemi istiyorum". My answer is I dont know. That is not the way we say it in Turkish. We say "Sinemaya gitmek istiyorum".

This is so much easier on my brain, makes perfect sense.

You´ve no idea how much I´ve struggled with the DIK suffix.

I have so many questions still, but I´ll save them for later  

Oh my Godness! Really! I´ve always thought that verbs with -me suffix are considered nouns... As there translation is ´my going´ which is a noun. 

Please correct me if I´m mistaken. 

"Ben neyi aradigimi aslinda bilmiyorum" VS "Ben aradigim neyi aslinda bilmiyorum"

Ben neyi aradığımı aslında bilmiyorum. (I know indeed what I am looking for.)

"Beni sevdigini biliyorum" VS "Sevdigin beni bilmiyorum"

Beni sevdiğini biliyorum. (I know you love me)

the patterns of -dik:

http://www.turkishclass.com/turkish_lesson_299 (the fundament of this topic.)

http://www.turkishclass.com/turkish_lesson_296

http://www.turkishclass.com/turkish_lesson_297 (your problem)

http://www.turkishclass.com/turkish_lesson_300

http://www.turkishclass.com/turkish_lesson_325

Benim gitmemi istiyor. (He wants me to go. / He wants that I go.)

This pattern depends on the -me/-ma suffix that one uses to make a verb a noun. But this -me/-ma suffix is not the negation for the verbs.

http://www.turkishclass.com/turkish_lesson_319

After you read this lessons, you may understand these:

git me (go ing) ------> gitme (going)

git me m (possessive) ---------> gitmem (that I go (purely : my going))

git me m i(accusative)---------> gitmemi (that I go [in accusative] : (puraly: my going [in accusative])

Gitmemi istiyor. (He wants me to go. (purely: He wants my going.))

/Benim gitmemi istiyor.

If you don´t still understand, say me, then I try to explain or give more example.

I didnt understand your question. Can you be more specific?