"It just really brought home that I did not walk alone, and it underpinned everything this club is about," he said of the Caseys.
While recuperating in hospital, Tim said he was overwhelmed by video messages of support from Kop heroes Robbie Fowler and Phil Thompson.
Tim said doctors had told him the Casey brothers had undoubtedly saved his life before paramedics arrived at the scene.
In an emotional reunion outside Anfield, Tim told Paul and Tony: "I will continue to cry my eyes out singing [You'll Never Walk Alone] for years to come, thanks to you guys."
"Everybody is in a rush to get somewhere," said Paul. "We stopped for five or 10 minutes and helped him, and he is here today.
"Had we not have stopped it would have been a totally different story."
Tony added: "Scousers help everyone.
"We're there to help everyone when we can, and Tim's here to prove it."
