느라

Because I was busy doing that

Says that being occupied with one thing caused something else — usually something that went wrong, got missed or wore you out: rushing out in the morning and leaving it behind, not noticing the meat burning while chatting, too busy reading the map, so busy preparing for a job. The two halves have to share the same subject, and the result is normally a cost rather than a benefit. 느라고 is the same thing spelled out in full, and plain 아서/어서 gives a reason without that flavour of being tied up.