This question has often plagued me as well. I don't have a definitive answer for you, but I might be able to free-write my way into a helpful explanation.
Sale, vale, and dale
Although I have never used
Dale in this group of contextual words, I'll just assume it extends this idea.
In English, when we understand something, or we want to say yes without saying
yes.. or pretty much anything at all, we say
Ahh.. Which are very similar to saying
yep, and so on. They all emphasize an affirmation. Often, such an affirmation is an indicator for the speaker to continue.
Speaker : I'm going to need 4 beers, 3 cups,....
Listener : Mmhmm (sale)
Speaker : 2 straws.... .....
Listener : ... uh-huh (vale)
Speaker : ....and 1 bottle of rum.
Listener : Gotcha! (dale)
The words can also be used to question someone's understanding
Speaker : In order to shoot the gun, it needs to have a bullet. Got it? (vale)
Speaker : The bullet is the key to shooting guns. Understand? (sale)
Speaker : After the next traffic light, make a left, Ok? (dale)
As far as the reason for using the verbs: valer, salir, and dar for the base of these words; I can only guess.
They most definitely have their regional usages, though I am unsure which ones are used where. I am not a native speaker, and I have only ever been to Mexico; other than my own country. While studying in Mexico, our teacher would often say
¿Sale vale? after one of her educational statements. I questioned her about it one time, while she was explaining the use of
dale... She answered saying that she wanted to be sure to use both to avoid any confusion... essentially translating her
sale vale into
Got it? Good!