One of the white wires is a neutral, and the other white wire is a "switched hot" coming from the wall switch. [EDIT: This was formerly an acceptable practice if the white was marked black, but is not now.]
(Of the two black wires one is the line hot (always hot) and the other black wire carries the always hot to one side of the wall switch.)
When a white is used as a switched hot (or an always hot) it is supposed to be marked with a ring of black tape to indicate that it is sometimes (or always) hot.
IMPORTANT FOR SAFETY
Notice that the two contact screws are different colors--one is brass (gold) colored and one is silver colored. When you connect the wires the neutral is to be connected to the silver colored screw and the switched hot to the brass colored screw. This specification is so that the switched hot is connected to the center contact in the screw base and the neutral is connected to the outer threaded section.
Before you remove the white wires from the existing fixture use a black marker on the white wire connected to the gold screw on the old fixture. Connect this marked wire to the gold screw on the new fixture. It would be best if you had a non contact voltage tester to confirm which wire is switched hot, but if you don't have one one could assume the existing fixture is wired correctly.