Argthtjtr

We've seen more than a few MCU characters visit Earth (or people from Earth talking with MCU characters off-Earth), but no one has (to my memory) had language issues.

When Thor first visits Earth, he has no problems in communication (culture shock, yes, but not communication). Carol Danvers in Captain Marvel makes a passing mention of a "Universal Translator", but I feel that this is humour rather than fact-based since I don't recall any other character make mention of this.

I don't believe that Tony Stark has invented a translation device because he may have mentioned it a few times.

Is the language issue explained, or is it simply ignored?

It is mostly hand-waving, but they actually do make references to universal translators. For example, Peter Quill is noted to have a universal translator implanted in his neck.

They seem to be common. They would almost be a necessity in a society where people can fly from one planet to another with such ease.

For Asgardians, they probably use something similar as "magic and technology are one and the same" to them.

EDIT

As Rarst pointed out in the comments, in the comics Asgardians have something called All-Tongue or All-Speak. This language gets translated to the native language of whoever hears it. This could be just another name for the "universal translator".

As far as I know, they don't mention All-Tongue in the movies, but they presumably have it.

In addition to DeeVs answer, it's mentioned in Infinity War that Thor took an elective at school to learn Groot so I would assume that it is taught at Asgardian schools since Groot is quite an obscure language as laid out by the films.