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?"