(I’m sure you can get a more explicit answer from someone else, but…)
– sometimes it just makes a statement more emphatic // emotional
– sometimes it is used to “indirectly express desires”
I think that first case is relatively easy to pick up on – really, I think if you just see it once, you get the feeling. Pay attention in anime or j-dramas for it ^^
As for the second one, I recall some situation like this from my Japanese class a few years ago:
Say two people are sitting together in a room and a window is open. If person A says 寒いだ～ without such emphatic markers, it’s simply acknowledging that it’s cold. On the other hand, if emphatic markers are used さむいんだ（わ・ね・な・じゃんetc), the feeling is more along the lines of “It’s cold (so maybe we could close the window?”.
To take a scene from real life (that I guess might have been caused by something else also)… a Japanese friend had lent me a book and we were walking to my dorm after lunch so that I could return it to him. As we got to my door I commented, ちょっと汚いんだけど。。 (It’s a little messy…) – to me it felt like a commonplace thing to say… but he actually stopped in his tracks a few feet away from the door, didn’t come inside, and waited (like 5 seconds) for me to fetch the book and come out. I hadn’t noticed that he had stopped because I was only inside so briefly, and laughed when I saw him waiting, asking why he didn’t come inside (as we were quite good friends). He responded that he thought I didn’t want him to/that it would be better not to enter (presumably because of the んだけど there)
Hopefully that’s clearer than mud :;^^