The England midfielder, 24, made his debut from the bench in Saturday's 1-0 home win against West Bromwich Albion.
Wilshere played only three games for the Gunners last season after fracturing his leg - the latest setback in a career plagued by injuries.
"If there is any question mark about me, I know it's about my fitness," he told BBC Match of the Day.
"Hopefully I can play as many games as possible, prove everyone wrong and then [reignite] my career."
Arsenal manager Arsene Wenger says Wilshere does not need to prove his quality as a player, as he considers him "world class" and wants the midfielder to spend his entire career with the Gunners.
Wilshere, who has started just 80 Premier League games for Arsenal in eight years, made two substitute appearances for the Gunners this season before joining Bournemouth.
Wilshere has 34 international caps and featured six times for England during the summer - including three games at Euro 2016 - but played no part in the 1-0 win over Slovakia in their World Cup qualifier earlier this month after being overlooked for new manager Sam Allardyce's first squad.
Bournemouth manager Eddie Howe says Wilshere's workload is being managed and insists he has signed a player who remains "the very best in his position".
Wilshere added: "I was happy to come on and was sitting on the bench watching and getting used to the way they play.
"It's similar to the way we played at Arsenal. They like to keep the ball and I can fit into this team, I think."
Jack Wilshere says his loan move from Arsenal to Bournemouth is about proving he can stay free of injuries.