Translation:Finally he said he accepts that he is forty years old.
Sure! I am going to start from the beginning of the word though for people reading this in the future. :)
Take the verb root and add the participle suffix: ol + duk = olduk
Now you have to add a possessive suffix for it to really have any meaning. In this case, the subject of the embedded clause is an understood o (which would have the form of onun if you chose to write it, which is 3rd person singular). So: olduk + -sI = olduğu
This is the direct object of the verb "kabul etmek" however, which means that you need the accusative (you are accepting a certain specific thing after all). When you add any case to something ending with the 3rd person possessive suffix, you always get a buffer letter -n.
This means: olduğu + -n + *-(y)I = olduğunu