Gennaro Gattuso’s side are fifth in the Italian top-flight and are in contention to return to the elite European competition for the first time since the 2013-14 campaign.
They were booted out of the Europa League group stage based on goal difference after a 3-1 defeat to Olympiacos in the final group game on December 13.
Following their exit from the continental stage, the Rossoneri ended a five-game winless run with a 2-1 home victory over SPAL before the winter break.
“Playing at San Siro gives you great emotions. When you think that 70,000-80,000 fans are watching you, it makes an impression on you,” Kessie said per Calciomercato.
“It has an incredible history and you feel it when you play there. San Siro isn’t like other stadiums, it’s different.
“We set the objective of qualifying for the Champions League at the start of the season, and we have to do everything to make it happen.”
Kessie who is on a two-year loan at Milan from Bologna has played in every of his side's league games this campaign and has hailed Gennaro Gattuso as a great coach despite recent pressure regarding his future at the club.
“Gattuso is a great coach. In general, when you play, you always like the coach,” he added.
“But he is an honest guy. When it's time to work, you work. With him, you have to be concentrated if you do not give the maximum, he screams at you.
“Once the work is done, however, he leaves you free to do whatever you want.”
The 21-year-old also reserved praise for teammate Gonzalo Higuain despite his struggles in front of goal this term and believes he can guide the club to achieving their objectives.
"Higuain? He is a great champion, he is very strong and a professional. We believe in him so much, we know he can take us to the top.”