The 22-year-old put pen to paper on a new four-year contract on Tuesday after breaking into the first-team picture under Jurgen Klopp last month following his recall from a loan at Swindon Town.
The midfielder, who joined the Merseyside club on an initial two-year deal as a free transfer in 2014 after being released by Tottenham Hotspur, made his Premier League debut as a substitute in the 6-0 thrashing of Aston Villa in the Reds' last match.
Stewart admitted that it was a huge boost to his confidence to be offered a long-term deal at the Premier League club, and set his sights on establishing himself fully at Anfield.
He told the Echo: "It gives me more confidence, it shows [the club] believes in me and believe that I could go on to become a good player for them.
"It took me until the age of 22 to make my debut and all those years I was just wondering what it’d be like, how difficult it would be, and if I’d cope playing in front of so many fans against Premier League players.
"[The contract] has just given me confidence in myself and the security [that I have] time to prove myself and get better with good coaches and good players."
Stewart, who never made a competitive senior appearance for Spurs, played in all four of Liverpool's FA Cup ties before making his top-flight bow at Villa Park.
He is currently ruled out of action for up to a month with a ligament injury, but will hope to continue his rise to prominence when he returns to full fitness.