A good way of thinking about it, in general, is that "spiller" is used with objects that doesn't necessarily have to be neither fun nor boring, such as a musical instrument (guitar, saxophone, trombone, etc.), board games (chess, cards, Monopoly). Compare it with playing an instrument or playing a game. It could be something that is challenging. "Leger" on the other hand is what kids do for fun, with or without an object. "Go out and play", "I'm playing with my dolls", "he's playing in the sandbox".
Spiller also has to carry something (usually a noun) with it. You can't say "Han spiller." It just sounds abrupt. Leger on the other hand is okay, you can definitely say "Han leger." It's the equivalence of just saying "He's playing".
Yeah, just to clarify this. "spiller" is a transitive verb, it must take an object. "leger" is not a transitive verb, if it does take an object, it usually requires a preposition, eg "leger med (noun)".
And as an aside, "leger" is one of the twin origins of the name "Lego" which, as well as being a latin verb meaning to play, is a contraction of the imperative "leg godt" (play well).
Hmm... well, "lege" means to participate in "leg" (noun) which means "underholdende aktivitet som især børn organiseret eller spontant udfører eller deltager i for fornøjelsens skyld", which translates to "entertaining activity which in particular children, in an organized way or spontaneously, perform or participate in for the sake of pleasure".
I think the key here is "in particular children", and generally it would be used for adult activities.
"Spiller" is used when someone is either playing a musical instrument or a game WITH RULES, like football or chess. "Leger" is used when someone is playing WITHOUT RULES, for example just playing with toys, or when someone is tinkering with something, learning how something works, for example electronics.
I hear it pronounced normally here, but the "g" sound can become contracted when followed by consonants, since Danish tends to be spoken fast. More commonly in Danish dialects, you will hear the "j" being dropped at the beginning and it can sound more like "eg" or even "æ" - particularly in South Jylland. Even in rigsdansk, there can be a tendency to drop the "j" when it comes after a verb, eg "hvor bor jeg?" can sound like "hvor bor æh?"