The Cherries agreed a deal with the League Two outfit on Thursday and the 24-year-old has now passed a medical and penned a three-year deal for an undisclosed fee.
"I'm happy that I've signed here and happy to be here," Grabban told the club's official website.
"We've been going back and forth with the club and my agent, and then I came back off holiday and was able to come and talk with Shaun Brooks. We were able to agree terms in a couple of hours and it was quite swift.
"I've played in League One before so I know a bit about it. I had a good season last season and I just want carry on the form I was in and progress again."