Arsenal had taken an early lead through Olivier Giroud, but Christian Benteke's brace and Antonio Luna's goal condemned the Gunners to an opening day loss.
There were chants from fans for Arsene Wenger to spend money after a quiet summer in the transfer market, with Yaya Sanogo the only arrival to date, and that was on a free transfer.
The Gunners face Fenerbahce in the first leg of their UEFA Champions League play-off in midweek and Wilshere is demanding an improvement.
"It's understandable," Wilshere told Sky Sports regarding the fans' reaction. "We have to be winning home games, there's no other way round it.
"We have to win games. The fans pay their money to come and watch us, and we have to put in better performances than that.
"We have to believe and we've got to go again. It's a tough season, a long season. We have lost the opening game, which is not ideal, but the good thing about football is we have a week to turn it around.
"There is a big qualifier in the week, and then a tough game away to Fulham, so we have to go again."
Arsene Wenger was unhappy with the second penalty Villa were awarded after Laurent Koscielny was adjudged to have brought down Gabriel Agbonlahor, and Wilshere also felt his team-mate had won the ball.
The midfielder added: "The second one, from my perspective, it was a good tackle, and he got the ball.
"The referee has to make the decisions, he's the one with the whistle. They'll even themselves out over the season."