How could they do that without explaining?
To me, this might also mean: Well, how could they dare do that without explaining? Well how dare they do that without explaining!
しれっと - as if nothing happened いてられる - is some Kansai dialect thing for いる
How they go together grammatically I don't know for sure, but I'll just have to take it at face value and just memorize this phrase as the translations I provided above. Often times I will just remember a set phrase as is without getting too technical in it, although I always like to dissect the grammar whenever possible.