Kjaer's loan from Sevilla became permanent earlier this week.
“Milan is simply Milan. It has always been a special club for me," he told Goal.com.
“Since I went abroad as a young player 12 years ago and arrived in Palermo, this was simply the club for me.
“Back then I told my agent that one day I would love to play in that red and black jersey.
“It has been close a few times before, but now the timing was right. Milan is a gigantic club with a gigantic history.
“I honestly believe that it is a privilege to play here and at the same time have the opportunity to help bring this club back to where we belong: to the top of Italian and European football.
“Since I arrived in January, we have been working hard, developing as a team, and I have seen great progress.
“We have a very young team with a lot of potential, and I believe that here in 2020 we have shown the way back to the top.
“It will demand a great deal of work from everybody, but I believe in a bright future for AC Milan.
“We have the history, the winning mentality, the passionate support of our fans, and great ambitions."