The 30-year-old flanker turned down a contract extension in Yorkshire for a return to Scotland next season. New Zealand-born Mather won the last of his four international caps three years ago - before he left Edinburgh for a season with Worcester.
Mather has enjoyed two campaigns at Leeds and hopes the experience he has gained down south can serve him well.
"Playing at Premiership week-in, week-out has given me the necessary competition to push myself and if I can bring back half of what I've learned at Leeds back to Glasgow then we can all benefit," Mather told The Herald.
"I didn't think there was a prospect of an offer from one of the professional teams, but David Jordan heard I was coming back to Scotland and made an approach," explained Mather.
"Glasgow's offer was not as good as the one from Leeds," continued Mather, who had already decided to move to be closer to his girlfriend before Glasgow made their approach.
"If it does help my Scotland chances then you can't put a price on that. However, my priority now is helping Glasgow get back to winning ways and if that leads to anything else then it's a bonus. I played in the last World Cup but it would mean an enormous amount to me to get into the squad this time."