Why does the Oxymercuration Demercuration mechanism lead to stereorandomization?
The stereorandomization occurs in the formation of the cyclic mercurinium ion intermediate. 



When the mercury acetate approaches the alkene, it can do so from either the top face:



or the bottom face:



Since the initial alkene is achiral and the steric factors are the same for both approaches, we get both ions, which are enantiomers.

Each enantiomer reacts stereospecifically with water to produce a chiral product. Note that the position where the mercury sits is still a chirality center in this example, but in many simpler cases it is not.





Finally, $\ce{NaBH4}$ is added to this compound as a second step to reduce the $\ce{C-Hg}$ bond to a $\ce{C-H}$ bond. In most cases, if the mercury was at a chirality center, that position no longer is one because we have two $\ce{C-H}$ bonds present.




  Why do we use mercury anyway?
  It works. It works to put the OH on the more substituted position without rearrangements typical of carbocations.


However, there is ongoing research to replace toxic mercury.