That's the work of Sam Manchester, deputy editor on the Sports desk and chief Photoshop wizard. I believe most of it is just cloning different parts of the photo to cover up the ball, though it can get more complex. For example, on the 4th photo of this previous round:

http://projects.nytimes.com/interactive/sports/worldcup/spot...

he actually cloned a players face from a different photo and pasted it in to cover up the ball. I have no idea how he does it so well.

I was wondering how he did that one!!! Part of my thinking was that you'd choose pictures that had the ball in an easy to photoshop location, and that threw me off

I think he enjoys a challenge. Case in point, the last photo in this set:

http://projects.nytimes.com/interactive/sports/worldcup/spot...

Maybe there's a way I just can't find, but it would be handy to be able to toggle everything 'off' again to get back to the original ball-less image.

Sports photographers also take many shots in rapid succession. I bet you could clone the background from a shot a half second before the one in the article.

If you look at the third picture, the guy in green is cloned and blended in where the ball is meant to be.

Suspect they used Photoshop Content Aware Fill. http://www.photoshopessentials.com/photo-editing/content-awa...