Arsenal announced on Friday that their club captain had agreed a new deal tying him to the club until 2028.
"It just shows what we're doing as a club," Ødegaard told reporters. "People want to be here as part of the project. What we're doing is really special and the players see what the manager wants to do and how the club is progressing, the connection we now have with the fans.
"I think it's a no-brainer for the players to be part of that. With so many hungry young players, it's a great place to be.
"You see the players around you want to be a part of what we're doing and it shows just how connected we are as a team. All the players want the same thing, they want to stay for a long time and really fight for the club and win things together. It shows the family feeling we have.
"Of course, I am thinking about [winning trophies]. That is the big dream, the big goal, to win things with the club. That's why I have signed here as well, because I believe we can achieve some great things. That's the big goal and I am sure if we work hard, we can get to something really nice."
Ødegaard said he believes Mikel Arteta's young side is ready to embrace the increased expectation that comes with finishing second in the Premier League last season, before spending over £200 million ($245m) in the summer market including the club-record £105m signing of Declan Rice from West Ham.
"It's a natural thing in football," Ødegaard said. "Expectations always go up if you perform. I personally enjoy that pressure. I like to have it. No matter the pressure from outside, I will always put more on myself.
"I think the team all enjoy the pressure. We want to be there at the top, fighting to win things. That's where we're going to be so we have to enjoy that pressure, focus on the right things and I'm sure good things will come.
"Even though we're a young team, you can see the quality, the responsibility the players take. Experience is not just about age. It's also about what you've done. I think the young players we have already have a lot of experience. Of course we need to improve all the time. But I'm sure with the mentality of all the team, the players, coaching staff, I'm sure we're just going to get better and better, and I want to be part of that."
The 24-year-old joined Arsenal initially on another loan in January 2021 but made a permanent £30m move later that year and has never looked back, scoring 25 goals in 92 appearances.
"In Madrid it was just a bit different. I was very young, I was stuck between the first and second teams, then I was moving around on loans here and there. So I never really got the time to settle down," he said.
"At Arsenal I've always had this special feeling. Even before I came here, I remember playing FIFA and I was always Arsenal. I had a good feeling about the club and I brought myself here.
"Just from the first day I felt this was a good place to be. With Mikel of course, after I spoke to him, what he wanted to do with the team and the club, how everyone is working together for the same thing. So it's a good feeling to stay longer.
"I didn't doubt myself [at Real Madrid], but of course there are moments when you maybe lose your confidence a little bit and you're not in your best place and don't perform as you want to.
"But I always kept that self belief and I was always sure I was going to turn things around.
"You have to be good at not focusing too much on the outside world. You have to focus on yourself, keep working hard and doing your thing.
"I think hard work always pays off. So I was pretty calm. Of course at some moments I was a bit disappointed, but I was calm and just kept on going."