Bairns captain Barr signed a pre-contract with Hearts in January and was desperate to move to Tynecastle having secured his current club's SPL status.
It was not to be, as Falkirk drew 0-0 at Kilmarnock on Saturday to drop into the Irn-Bru First Division.
Scotland defender Barr, who has spent his entire career with the Bairns, told the club's website: "This isn't how I wanted to leave things. It's hard to take and it's even harder to talk about it.
"It's important the club bounces back as quickly as possible. No one wants to be in Division One instead of the SPL, but some good can come of it.
"Hibs seemed to come out of a season at that level in better shape and Inverness Caley have re-emerged stronger this year.
"We've got great staff that work really hard behind the scenes and there's the nucleus of a good young team. I've no doubt the lads will kick on and get back up, definitely.
"I was desperate to leave on a better note and it's been hard to accept that those were my last few days as a Falkirk player.
"It was tough enough clearing my stuff out of the training ground and saying goodbyes to the staff on Friday, but I suppose it's part and parcel of football.
"You just build up a great attachment to the club after so many great years and it's been one amazing journey over the last 10 years. This has been my club and it will always be close to me.
"Obviously, the hurt's still raw from Saturday, but there have been so many highlights for me.
"This was the club that gave me my chance at being a footballer in the first place, allowed me to play alongside quality players like Russell Latapy, play in Europe and win a Scotland cap.
"There was the Scottish Cup run and, until the last two years, there have been some great league campaigns.
"On the flip side, doing my cruciate ligaments in my first year as a full-time player at the club at 17 was hard to take. I had 12 months then to prove myself and to win another professional contract.
"Basically, what I'm trying to say is that I'm really grateful for everything Falkirk have done for me and the support I've had over the years.
"I've so many people to thank that I don't want to list them for fear of missing someone out. They all know who they are."
Source: Team Talk
Source: Team Talk