Put most simply sodium quite easily gives up its outer most electron and becomes the positive sodium ion. What they found was that when sodium is put into water the water sucks the electrons from the surface of the sodium metal. Without the electron you just have positive ions which repel each other this causes small needle like fingers of sodium to extend into the water. This greatly increases the surface area causing hydrogen gas build up and allows for the big explosion.
The high speed images show the needle like filaments. The crucial frame is 0.4ps 5th image from the top middle column where you can see the needles of sodium and potassium (potassium is added to make a liquid metal for easier dropping). The next frame shows the explosion at the surface.
Now comes the awesome simulations. So using quantum calculations they modeled nano drops and show the electrons (blue in the image below) moving into the water and then the sodium ions breaking apart as there are no longer any electron glue to hold them together.
Scaling up the simulations using some assumptions that reduce the amount of computer processing time they showed a larger drop exploding at the surface.
Check out the movie they put together and the paper. I wonder what other elementary reactions are still not fully understood.
Mason, P. E.; Uhlig, F.; Vaněk, V.; Buttersack, T.; Bauerecker, S.; Jungwirth, P. Nat Chem 2015, advance online publication.