The 25-year-old has agreed a three-year deal with the European champions from the start of the 2017-18 season.
The switch - announced last month - will end a six-year association with the Scarlets which has seen Williams score 26 tries in 102 appearances.
"It was an opportunity I couldn't turn down," said Williams, who lines up for Wales against Scotland on Saturday.
"Hopefully, I'll go there and be part of a side that wins a couple more trophies.
"Saracens are European champions and were in another final two or three years ago. They've been English champions for a couple of years running."
The 40-time capped wing or full-back added that living closer to his girlfriend Sophie Harries, a London-based model, was another factor in his decision.
"It's a new chapter. I'm ready to try something different and looking forward to it," he added.
"I still have some games left for the Scarlets. We're looking to make the top four in the Pro12 and get into the play-offs and I want to help them achieve that.
"It would be brilliant if we could win a trophy this season."
Williams has not given up on adding the 2017 Six Nations title to his trophy cabinet.
The 21-16 defeat by England leaves Wales third in the table on five points, but Williams believes a win against Scotland - also on five points - could rekindle their challenge.
After Scotland, Wales play Ireland at home before rounding off their tournament with a trip to France.
"We have three tough games to play," said Williams, who has scored against Italy and England.
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"We have to get our performances right and make sure we win our next three games. There's still a chance we could win the championship."
Williams' versatility in the back three make him a potential candidate to tour with the British and Irish Lions in the eyes of some pundits.
"It's always been a dream of mine to make a Lions tour," added Williams.
"It would be the pinnacle if I could get there, but I've learned during my career you should never look too far ahead.
"So all I'm concerned about at this point is the Six Nations."
Williams has played most of his international rugby on the wing despite being a full-back for Scarlets.
But any rivalry with Wales and 2013 Lions number 15 Leigh Halfpenny is in the minds of fans, with Williams saying he and the Toulon player are on the best of terms.
"I prefer to play at 15 but Leigh is doing an immense job and so I'm happy to slot in on the left wing or the right wing - anywhere in the back three," he said.
"We are good mates. At the weekend his local team, Gorseinon, played my local side, Waunarlwydd, and Waun won with the last kick of the game, so I have the bragging rights."

The 31-year-old lock has made 62 international Test appearances, playing at both the 2011 and 2015 World Cups.
Nemsadze has also played for Montauban and Valence d'Agen among others.
Bristol director of rugby Andy Robinson said Nemsadze, at 6ft 5ins and weighing more than 120kg, "fits the bill for a Premiership lock".
Robinson continued: "He's heavy, abrasive and he's been a key part of a Georgia forward pack that has enjoyed considerable success at international level.
"We recruit players that fit the mould of the standards and culture we expect at Bristol Rugby. Giorgi not only brings with him the knowledge of playing in France for nearly 10 years, but also that physical intensity we are striving for."
Wales back Liam Williams hopes to win "a couple more trophies" after his move to Saracens in the summer.
[NEXT_CONCEPT]
Premiership newcomers Bristol have signed Georgia international lock Giorgi Nemsadze from French second-tier side Tarbes.