Y is it that so many groups from other countries sing their songs with no trace of an accent?

I'm not sure you can generalize but I think most singers who sing in a second language practice their diction pretty carefully. Abba sang not just in English, but also in Spanish, in addition to their native Swedish. They even recorded a few tracks in French and German. I think they were very careful to learn their lyrics carefully and, since they're singing, the musical notes substitute for the natural rythym of the language.

I assume you were watching Benny Andersson and Bjorn Ulvaeus speak. Their English is pretty excellent, especially Bjorn, who lived in London for many years. Still, during their recordings, there are still some traces of Swedish accent that many find adorable. For example, listen to their track Happy New Year. Agnetha, who was never confident speaking English, clearly says champ/pagne, with a hard 'p' sound. You can hear little touches of accent like that throughout their songs but it just adds to the charm, in my opinion.