The Premier League champions were the last surviving English team in this season's competition.
But, despite a spirited second-leg display, a 1-1 draw meant Atletico progressed via a 2-1 aggregate win.
"The whole club, from the supporters to the players to the owners can be immensely proud," said Shakespeare.
"We had them rattled with the effort and commitment we showed. It's no discredit to lose to a team of that calibre.
"I've just said to the players that they should want more of this and they've agreed that that's what they want.
"All players want to play at the highest level and the Champions League is the highest level but we have to get back to winning ways in the Premier League now."
Seeking to overturn a 1-0 first-leg deficit, Leicester fell further behind to Saul Niguez's header in the first half on Tuesday, meaning they required three goals to progress.
The Foxes dominated the second half and gave themselves hope when Jamie Vardy finished from close range on the hour mark.
But, despite sending on Leonardo Ulloa for Shinji Okazaki at half-time and leading the shot count 17-2 after the break, they were unable to breach a resolute Atletico defence for a second time.
"In the first half, we played really well but the goal changes the game plan - we knew we had to score three - so I had to make the change," Shakespeare told BT Sport.
"In terms of effort, commitment, application - as a group we were tremendous.
"The momentum was with us when Jamie scored but it just wasn't to be."
Shakespeare was appointed Leicester manager until the end of the season following the sacking of Claudio Ranieri in February.
The 53-year-old, previously Ranieri's assistant, oversaw six wins in his first six games to steer the Foxes away from the relegation zone.
When asked about his future, he replied: "It's not in my hands. It's in the club's hands. I've said we'd sit down at the end of the season. I'm more than happy to do that before if it arises but the contract says until the end of the season.
"I've enjoyed it, pitting your wits against one of the best managers in the world, one of the best sides in the world.
"Now is a time to reflect and I'm sure at the end of the season I'll be able to reflect on my own performance as well as the club's."
Ex-England defender Danny Mills on BBC Radio 5 live:
"Ranieri wasn't on anybody's radar and clearly they went for a big name and it worked.
"There's just part of me that feels a little bit sorry for Craig Shakespeare because the same thing might happen again.
"He almost becomes unemployable because wherever he goes. If he goes in as a number two, the moment you have a bad game [people will think] he's going to want the job and someone's going to get sacked.
"So no-one is going to want to employ him as a number two. Does he get a big job as a number one? I think that will be difficult. Will he want to step down to the Championship? Probably not, because he's still very inexperienced."
Atletico Madrid have now reached the last four of the Champions League in three of the past four seasons.
"I'm full of emotions, full of pride for the performance of my team," said manager Diego Simeone. "Full of hope and excitement as we progress.
"But I have to say what a great performance from Leicester. It was almost a pleasure to compete against them. We were living in fear all night of what they might achieve. They pushed us all the way.
"We performed in the way this match needed us to perform. We always come up with a solution. I don't like to praise too much but we responded and played in the way we needed to."
Manager Craig Shakespeare challenged his Leicester team to reach the Champions League again after their quarter-final loss to Atletico Madrid.