I can give some insights into the why. The how probably was through the usual process of licensing foreign songs.
The anime was made by Madhouse who contributed two episodes to the 2003 anime Animatrix. This project was important at the time, since it was a direct result of the anime-craze in the west in the early 2000s. Anime studios in Japan started to pay attention to foreign markets and with people behind a major Hollywood success like The Matrix reaching out to Japan to create something like the Animatrix, chances with the western market seemed good.
And thus a lot of studios did some projects betting on that western market. For Madhouse, these projects included Texhnolyze, Monster and Gunslinger Girl, each of them featuring a song by a foreign artist and deviating from usual expectations of anime fans, who just discovered their fetish for maid characters in their moeblob harem shows, that casual western fans didn't even knew existed.
Gunslinger Girl was part of the studios efforts to appeal to the western market, which is why obscure foreign artists ended up in anime of that time. At least that's my attempt at explaining how this came to be. I don't have any inside knowledge and am speculating from an observer's point of view.
The second season came out five years later and was handled by a different studio. To be specific in 2008. It was at a time everyone betting on the western market realized that this was a huge mistake. Studio Gonzo who made the biggest bet was in the most miserable state and talks of bankruptcy went through the news.
It was under this circumstance, that the studio handling the second season tried to deemphasize the appeal to the western market and therefore changed the character design towards something I described above as moeblob. This is purely speculation on my part, but it seems like a valid assumption that this was the motivation behind the change.
It also didn't help very much, that the manga itself wasn't exactly going strong. Most of the story weaknesses were copied directly from the source. It just so happened that the first season made subtle changes in the presentation to appeal to western audiences, which the second season avoided.