しれーっといてられるなと思います


> 「まぁ、よくも、こんな重大{じゅうだい}な事件{じけん}を視聴者{しちょうしゃ}に説明{せつめい}もせず、しれーっといてられるなと思{おも}います。」

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.




As you already know, you can parse it as:

> 『まぁ、よくも、こんな重大な事件を視聴者に説明もせず、しれーっといてられるな。』と思います。

> しれっと - as if nothing happened  
>  いてられる - is some Kansai dialect thing for いる

Yes, you're basically correct.

(I first thought the いてられる should be a typo for して(い)られる but after some
research I found in [this video](https://youtu.be/iRi4od_Thus?t=169) the
speaker clearly says しれーっと、いてられるな. According to
[Wikipedia](https://ja.wikipedia.org/wiki/%E7%AB%8B%E8%8A%B1%E5%AD%9D%E5%BF%97#%E6%9D%A5%E6%AD%B4)
he was born and grew up in Osaka.)

「しれっと」,「しれーっと」 or 「シレっと」 means 「何事もなかったかのように、平然としているさま」 (『明鏡国語辞典』),
"nonchalantly" "as if nothing happened".

いてられる here is Kansai dialect for いられる(居られる) "can be (there)" "can stay", which
is the potential form of いる. Here in Kansai, especially in Osaka, we say
いてる(居てる) to mean いる(居る).

You're also right that 善くも means "How can you have the nerve".  
よくも is often used with a potential form:  
「よくも～～られるな！」-- "How can you/they have the nerve to do...!" "How dare
you/they...!"

So, よくも[...]しれーっといてられるな、と思います here literally means:

> I think "How can they have the nerve to stay, as if nothing had happened."

