The 30-year-old Republic of Ireland midfielder, who will join the Portman Road club until January, said that the need for first-team football was behind the switch.
"I didn't want to be in and out of the team this year and I need to be playing football so I can feature for my country," he told BBC Radio Suffolk.
"I don't see this as a step down at all. I see myself coming here and hopefully making a good impression."
Rovers signed Andrews from MK Dons in 2008 for around £1m.