Here's the sentence in question:
生まれたのも育ったのも京都なんですよ
I thought using の in a sentence signified possession (e.g. 私の本 [My book]). If I interpret the sentence as such, it doesn't make any sense.
Then I came to the conclusion that I should include the も from the sentence and read it as のも, but doing that made the sentence even more confusing.
In linguistics, there's something called a cleft sentence. The basic idea is that you split a sentence into two parts in order to focus something:
１ａ. I met her that day. (original sentence)
１ｂ. It was her that [I met that day]. (clefted sentence)
In this example, we split the sentence into "I met __ that day" and "her". Because we've neatly divided the sentence in two, we call this a "cleft" sentence. It expresses the same basic information, but it puts the focus on "her", the element we pulled out.
We could just as easily focus a different element instead:
２ａ. I met her that day. (original sentence)
２ｂ. It was that day that [I met her]. (clefted sentence)
All of these sentences express the same basic information, but their focus is a little bit different. We could think of our first cleft as answering the question "Who did I meet that day?" and our second cleft as answering the question "When did I meet her?"
With that in mind, let's look at our examples again:
1ｂ. It was her that [I met that day].
non-focused element: I met (𝑥) that day focused element: (𝑥) = "her"２ｂ. It was that day that [I met her].
non-focused element: I met her (𝑥) focused element: (𝑥) = "that day"
The non-focused element introduces a variable 𝑥, and the focused element tells us the value of 𝑥.
Japanese has cleft sentences that work like this, too! You've probably seen them before, although you might not have known the technical term. The most basic type of cleft looks like ＡのはＢだ in Japanese, where Ａ introduces a variable 𝑥, and Ｂ tells us the value of that variable.
Let's use a simplified version of your sentence as an example:
３ａ. [京都]{﹅・﹅}で生まれた。 '[I] was born in Kyōto.'
This is a basic sentence. We want to turn it into a cleft sentence, so let's split it into two parts!
３ｂ. 生まれたのは[京都]{﹅・﹅}だ。 'It's [in] Kyōto that [I] was born.'
non-focused element: (𝑥) 生まれた focused element: 京都で = (𝑥)
We want to focus [京都]{﹅・﹅}で, which goes at the end of the sentence before だ (dropping the case marker で). Then we add the nominalizer の to the non-focused portion, 生まれた, so we can make it into the topic of the sentence.
Now we have a statement containing a variable ("[I] was born 𝑥") and we also have the value of that variable (𝑥 = "[in] Kyōto"). Note that the final sentence is missing the case particle で, so we have to understand the relationship between the two halves from context.
By the way, sometimes it's natural to use a cleft in Japanese when we'd use a non-cleft in English. So don't get too hung up on the English translations here―they're designed to show you the meaning of the Japanese sentences, not to give you the most natural way to communicate in English.
As it happens, your sentence is a little more complicated! To come up with your sentence, we'll need to put two of these things together, and of course when we do so we'll naturally need to use も instead of は. Let's start with these sentences:
４ａ. 生まれたのは[京都]{﹅・﹅}だ。 'It's [in] Kyōto that [I] was born.'
４ｂ. 育ったのは[京都]{﹅・﹅}だ。 'It's [in] Kyōto that [I] was raised.'
And combine them into this:
４ｃ. 生まれたのも育ったのも[京都]{﹅・﹅}だ。 'It's [in] Kyōto that [I] was born and raised.'
Finally, to get your sentence we have to make a few more changes. We'll add のだ, contract の to the informal ん, replace だ with the polite です, and add the assertive final particle よ:
４ｄ. 生まれたのも育ったのも[京都]{﹅・﹅}なのだ。
４ｅ. 生まれたのも育ったのも[京都]{﹅・﹅}なんだ。
４ｆ. 生まれたのも育ったのも[京都]{﹅・﹅}なんです。
４ｇ. 生まれたのも育ったのも[京都]{﹅・﹅}なんですよ。
Since the exact meaning of のだ depends on context, I've provided no explanation here―that could, of course, be the subject of another question.
And that's it! For more explanation of cleft sentences, see Martin's 1975 Reference Grammar of Japanese, p.863-869. For more explanation of English clefts, see Huddleston & Pullum's 2002 Cambridge Grammar of the English Language, p.1414-1427.
alexhatesmil's answer isn't wrong, but I just want to supplement regarding the も and what it's doing.
So first, like the above の is a nominalizer.
But that means you need something to connect の blocks to the rest of the sentence. You can do so with either は or も
so you could say:
育ったのは京都です。
= The place I was raised is Kyoto.
The も here replaces は as a topicalizer and adds "as well as"
生まれたのも育ったのも京都です。
の is being used as a nominalizer (something that turns a verb into a noun) in this case, with も meaning "as well as". So, literally, it would read something like "My being born as well as my growing up were in Kyoto." In other words, "I was born and raised in Kyoto."
