The 23-year-old, whose transfer terms remain undisclosed, ends a 10-year association with the Cherries, having come through the club's age groups.
He made his full debut in August 2015, one of only two appearances, but did have loan spells at Yeovil, Woking and last season, St Johnstone.
"Matt is someone we tried to sign last season," boss John Coleman said.
"So to get him on a permanent deal is a big coup for us. He is a talented player, a big presence in midfield and I am looking forward to working with him."
Butcher added: "I am raring to go, my loan spell at St Johnstone was cut short by Covid and I want to get back into the swing of things as quickly as possible."