Learn English – What does the phrase the more you know mean

Just so you know is a pretty ordinary construction, as such things go in English.

Just = "only"
So is the ordinary adverb, employed in the sense "in order that"
*You know" = "you know", "you are aware of"

So:

I provide this information (without being asked) in order that you may be aware of it and thus be spared embarrassment or inadvertent error.

Both questions are asking how an individual acquired knowledge about the apples and can be used interchangeably in most circumstances. There is however a slight difference in nuance.

How did you come to know? emphasises questioning the mechanism by which the information was obtained. Did you search for the information, stumble across it or were given it? It also carries an implicit question of why did you want to know?

How do you know? still questions the mechanism and, as you say, carries the same basic meaning, but has no implicit why.