Why is the article omitted?
In Dutch, 'uitleg geven' is considered the infinitive (Just like in 'lesgeven: to teach) . In a sentence however two halves don't have to stay together. they do not require an article.
Still, while less charming, I would consider 'Ik geef een uitleg' also right. It applies to a more specific explanation if there are several.
True. I'm not saying you should omit the article. It just isn't required, which was the original question.
Then shouldn't this be "I explain"?
Not really, because that would be "Ik leg uit" in Dutch.
However, for now, it is an accepted alternative. :)