So, it is my understanding that 日の出 means sunrise and 日の入り means sunset. However I'm having difficulty understanding why. 日の出 to me would mean, "The sun's exit", and 日の入り to me would mean "The sun's entrance". Can anyone help me better break down these nouns? Is there some meaning to 出 and 入り that I am missing?
The picture here is actually about the horizon (地平線).
日の出 means that the sun "comes out" from the horizon.
日の入り means that the sun "goes into" the horizon.
Hope this could be helpful to you.
Well, you're at the very least on the right track. I think your confusion is coming with the stem of the verb 出る. Now, you're right in that it can mean "to exit", but a better, broader meaning would be something like "to come out", which would apply to 日の出. For that one, think of it more like, "the coming out of the sun (from the horizon)", which is closer to the meaning of "sunrise" is has.
The same goes for 日の入り. 入る means something more like "to enter" or "to go into" in a broader sense. What this means for you, however, is that you can think of 日の入り as meaning "entrance of the sun (into the horizon)".
Isn't the concept (of how sun behaves) natural even for English speakers?
日の出 the sun comes out
日の入り the sun sets in
The only subtle difference being the verb "set in" instead of "go in", but the direction is preserved.