ben en çok seni anneni ve aileni üzdüm diye kalbim aðrýyor...........

 I am the most sad for your mother and family,my heart is aching....

ben en çok seni anneni ve aileni üzdüm diye kalbim aðrýyor...........

The ´diye´ should be translated as ´because´

It then becomes: 

My heart aches because i´m sad, mostly about your mother and your family

diye can also mean ´so that´

 I am the most sad for your mother and family so that my heart is aching....

The ´diye´ could be translated with ´so that´ but only in the meaning of ´in order to´

I went to bed extra early so that I would get plenty of sleep.

I went to bed extra early in order to get plenty of sleep.‎

In the sentence

‎ I am the most sad for your mother and family so that my heart is aching....‎

the ´so that´ has the meaning of ´as a result of that´, and is therefor no good translation of ´diye´