1006 GMT June 24, 2018
The Premier League champion was the last surviving English team in this season's competition, BBC reported.
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 caliber.
"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 it 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 halftime and leading the shot count 17-2 after the break, they were unable to breach a resolute Atletico defense 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."
'We'll sit down at the end of the season'
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."