As someone who has edited an anthology, and as a writer who needs an editor, editing is extremely important.
I don’t agree with Sean’s comment that publishers will drop a submission after the third spelling mistake. That hasn’t been my experience with any of the publishers I have dealt with both in and outside of the fandom. I will say, however, that if there are a huge amount of errors, it will drastically lower your chance of being accepted into the anthology/being published.
Time and time again on podcasts such as Dead Robots Society, they mention that if you are going to independent or self pub, you need a professional editor, and they are worth every cent. I agree with this, but you also have to make sure the editor you are selecting is good at what they do. Just like any other business, there are those out there that just want to take your money for little to no work, so keep your eyes open.
But yes, all work needs to be edited by someone other than you. Be it during the beta reading phase or by a professional editor, it needs to be done.