Meaning of っていうのは without っていうことだ

I think nothing is omitted after 感じる. 感じるということんやろ is ungrammatical in the first place. But ん in 感じるんやろ is explanatory-no, the same ん in んだ/のだ.

Starting from this:

昔行った場所に久しぶりに行くから、懐かしいと感じる。
Because you go somewhere long after the last visit, you feel nostalgic.

Changing the word order for emphasis (cleft sentence):

懐かしいっていうのは昔行った場所に久しぶりに行くから感じる。
It's because you go somewhere long after the last visit that you feel nostalgic.
→ "Nostalgic" is something you feel because you go somewhere long after the first visit.

^{(Here っていうのは is basically an emphatic version of のは, but it's mandatory because bare 懐かしい cannot work as a subject.)}

Adding だって and んやろ (which is dialectal のだろう, explanatory-の + だろう):

だって、懐かしいっていうのは昔行った場所に久しぶりに行くから感じるんやろ。
But..."nostalgic" is something you feel because you go somewhere long after the last visit, isn't it?