Why are mice with a single X chromosome and no Y chromosome males?
Almost all mammals (including mice and humans) have two sexes where the males have a Y chromosome and an X chromosome (whereas females have two X and no Y chromosomes).  This is not the only way organisms can determine sexes, but it's the way mammals evolved to do it.

So how does having a Y cause an individual to be male?  The Y has only a few genes (less than all other chromosomes), but it has genes that occur only on the Y.  One of these in particular, the one called SRY, is the main gene that causes the individual to develop male characteristics.  It mostly does its job indirectly, by making a protein that affects many other genes on other chromosomes that in turn do the work of building and maintaining the male form.

In the experiments reported by the article mentioned in the question, they managed to cause mice without a Y chromosome to develop many male characteristics.  They did this by faking one or two of the actions of SRY.  One of the many actions of SRY is to turn on a gene called Sox9 (which is on a non-Y chromosome), and it turns out Sox9 causes most of the things that make the male mouse form.  

So the experimenters found a way to turn on Sox9 without using SRY or anything else from a Y chromosome.  This got them mice that looked male, but didn't produce sperm.  So they did something further that substituted for a Y chromosome, and caused a weak gene on the X to work at many-times strength to do the job that a strong Y gene normally does.  This caused the special "male" mice to produce sperm.  But these sperm were not fully functional (they couldn't swim and enter the egg), so they showed they were essentially sperm by simply using a tiny needle to inject the semi-functional sperm into mouse eggs and have those eggs develop into viable baby mice.

Bottom line, if you use artificial methods to do what the Y chromosome does, you can make male mice which do not have a Y chromosome (and learn some things in the process).