I really Schruted this one at first. Since I hear it from kids a lot (and my wife when she talks in a cute way) I thought it was a way of making the sentence sound cute. Like a built in grammar point for "Andy have a boo-boo tummy".
Instead, it gives the listener a reason for something. If the speaker is saying something that the listener doesn't know the reason behind the action, then this verb ending helps to clarify "this is why". It sounds cute when it's used cutely but it has a genuine meaning outside the baby world. It also is very context heavy so it's hard to give a single, one sentence explanation.
A: 왜 전화 안 했어?
B: 어? 전화 했다고?....못 들었거든.
A: 아~ 그랬구나.
A: Why didn't you call?
B: Huh? You called? I didn't hear it...
A: Oh, okay. No biggie.
Careful because the same conversation can sound pretty coarse depending on the intonation.
A: 왜 전화 안 했어?
B: 전화기 소리가 안들렸다고!! 나보고 어떡하라고!!!
A: 뭐? !@$@%$#%$#%
A: 왜 성당에서 결혼식을 올렸나요?
B: 신부가 천주교 신자이시거든요.
A: Why did you have the wedding at a cathedral?
B: Because the bride is Catholic (that's why)
A: 매튜형은 왜 지인이와 만날 때마다 다투나요?
B: 둘다 미국 사람이지만 나는 텍사스 출신이고 지인이는 캘리포니아 출신이거든요.
A: Why do you always bicker with Linda?
B: Both of us are Americans but I'm from Texas and she's from California (that's why).
"~거든" can feel friendly at times and cranky at other times. I didn't really leave the best examples but hopefully this will be enough to catch it in real life.