You misunderstand the role of pi. The statement refers to how many digits are needed to calculate the volume of the universe to a certain degree of accuracy:
For most numerical calculations involving π, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmological calculations, because that is the accuracy necessary to calculate the volume of the known universe with a precision of one atom.
The other answer disputes the statement, as do I. The footnote in Wikipedia, and a search for the statement, takes you here:
Here are my calculations for the size of the universe:
4.600E 10 radius in light years
9.461E 15 meters per light year
4.352E 26 radius in meters
2.734E 27 circumference
3.452E 80 volume
Here's what I get for an atom, based on the diameter of a hydrogen atom being about .25 Angstrom.
2.500E-11 diameter of hydrogen atom in meters
1.094E 38 ratio of circumference of universe to diameter of atom
8.181E-33 volume of a hydrogen atom.
4.220E 112 ratio of volume of universe to volume of atom
I believe that the original statement in Wikipedia was that 39 digits was sufficient to determine the circumference of the universe to within the diameter of an atom. The above calculations bears that out. For volume, you need about 113.
I don't know when the statement was changed to volume, but it appears you need pi accurate to about 113 digits to get the volume accurate to within the volume of an atom.
Here's a demonstration of the calculation by someone else:
http://gizmodo.com/5985858/how-many-digits-of-pi-do-you-really-need and it's basically the same as what I did.
I'll look into the history of the Wikipedia article and try to get it corrected. I've posted a comment on this in the to-do list for the Pi article in Wikipedia. It's a featured article, and I'm reluctant to contradict their quoted source without going through some owner or administrator of the article.
I'm still trying to figure out how a universe 14 billion years old has a radius of 46 billion light-years. I went to the article citing that, but it involves cosmological reckoning that are beyond me. But the video I referenced comes up with a diameter based on about the same radius.