Admit your mistake
Admit your mistake. That's the most important part.
Do not excuse it
Do not explain it. Do not justify it. Do not give out reasons.
There are exceptions to this rule. But for now, I won't go into them. 99% of the time, it's just better to not give out any.
Do not promise that you won't do it again
If you want people to trust you, ...
All of the other answers are good. But I think it's worth thinking about it another way as well.
How do you, as a leader, think a team member should behave if they miss a deadline? I think a leader should behave basically the way they WANT a team member to behave. And do it in front of the whole team.
How should he react now?
Since you said your friend a leader, the leader does not react...
Shall he give an explanation for the missing deadline?
Or just keep quiet and try to build a better reputation next time by being in-time?
An apology & an explanation is appreciated and will be seen as your courage for acceptance of failure. An ...
How should a leader behave when he misses deadlines himself?
Leaders need to be held accountable just like anyone else. If they missed a deadline they should:
Admit to missing the deadline
Apologize for missing the deadline
Explain the reason(s) for missing the deadline
Explain the steps that they will take to prevent this from happening in the future.