Can someone explain to me

That is, when a team has done something illegal, why does the play continue when that team is going to be penalised?

In no sport in the world does this occur.

That punt where the Seattle runner returned it to the Atlanta 9, the penalty occured when the Atlanta player kicked it. The ref who threw the flag obviously threw the flag because he saw the Seattle player did something illegal, so the penalty was going to be on Seattle....

The question is, why allow play to continue? Most stupid incompetent process in the sport.

In rugby, if say a team that has the ball does something illegal, as soon as it is done, the play immediately stops. It doesnt allow the play to continue then bring it back for that penalty.

Why bother letting them to continue to play if something was done illegally?

If a team has the ball and the opposition does something illegal, then absolutely play the advantage, but ONLY if it means the team that has the ball has the advantage.

I mean, the guy ran the ball like 70 yards or whatever yet his team mate did something illegal way way before, why not stop the play immediately instead of letting play continue to bring it back?

Most dumbest thing i have ever seen in sports.

It would be like if a basketball player does something like a travel, do the refs let them continue to play offense until they finish there offense set then bring the ball back?

You have a very good point. But there is also advantage for the team that was just fouled. If the punt was muffed & kicking team recovers it,they can decline that penalty & get the ball in favorable field position. So fouling team may get penalized more by committing the foul.

Returner could fumble.

On offense, if there is a hold on the offense the play continues and it could result in an interception for the defense.   

Isnt this number very low? In that case, is it worth it at the expense of speeding up the game and useless injuries?

And to answer the OP, there is also delayed penalties in hockey.  If a team commits a penalty without the puck then the play does not stop until they touch the puck.  This allows the team that has been fouled to pull the goalie and get a 6th skater on the ice to try and score before the opposing team touches the puck to stop the play.  Just gives an extra little advantage to the team that has been fouled to continue the play where they may have had a scoring chance.  Obviously the team getting the penalty can't take a shot at the empty net but every so often a bad pass or bounce can result in the team that's about to get a power play putting the puck into their own empty net.