The England international ended his association with the Gunners in January following over two decades at the club.
Walcott, 28, had become a peripheral figure in north London after slipping down the pecking order.
The Toffees offered him a Premier League lifeline and coughed up £20m to bring him to Merseyside.
Walcott says the lure of working with Allardyce helped him make up his mind.
“Sam was one of the main reasons why I came to the club,” he told Sky Sports.
“I wanted a manager like that who could get the best out of me.
“I had that short stint with Sam with England, and to see it now every day, the way the intensity of training is and how demanding it is, has been great.”