Williams was offered a three-year-deal at Crewe but the Manchester born keeper decided to sign for John Ward's Calisle United.
Although Williams made 49 appearances last season for the Railwaymen the player came in for some stick from some sections of the crowd.
Williams told a local newspaper 'I had a very good offer to stay at Crewe but I felt the time was right for me to seek a fresh challenge elsewhere and Carlisle fit the bill in every way.
"I need to progress my career and I`m sure I`ll do that with Carlisle.
"It helps as a keeper when you`ve got good defenders in front of you and there are some very good defenders at Carlisle. I aim to be a very reliable last line of that defence.
"I think [Westwood] kept 19 clean sheets for Carlisle last season but I kept 13 in a team that struggled throughout. I`m proud of that."