Trying to understand why a light would be wired this way?
So, the box in the ceiling has two pair coming into it.  Two black white pair.   One comes from the power source.  Think of that one as Line and Neutral.  The other one runs to the light switch wherever that is.  (  If you open the switch, you should find a black and white on the two screws.  )  

Therefore, the two black wires are connected and beep your meter because the breaker is still on.   Keep them intact.   AND the white coming from the switch is connected to the black on the fan.  It SHOULD have a bit of black E tape to mark it as a 'switched hot'.  The white neutral is connected to the white wire on the fan.

How'my doin' so far?

The new fixture should be wired just like the old fixture.   Put the black where the black was, and the white where the white was.   

Double check the box is secure enough to the house to hold the weight of the new fixture.