Hiding true ownership. You can give % of shares to a politician in exchange for a law that will directly benefit you. Politician will be able to monetize on this once he is out of office.
Anti-monopoly regulations. You can own two companies that on paper are competition but in reality are same entity blocking any other company from entering the market.
On the legal side there are actually probably more reasons, anonymity being one (I.e. Hiding from your neighbors who just bought the house next door), numerous legal tax avoidance reasons (two common ones would be: 1. Because of the u.s. Tax code (unlike most other places) the u.s. Charges corporate tax on foreign income but only when it is repatriated - so many companies will avoid tax by having offshore subsidiaries that hold IP to keep profits from coming back to the u.s. and being subject to tax (see every big software company with irish subsidiaries or swiss corporate inversions) 2. Hedge fund/private equity business makes extensive use of offshore companies so that non-u.s investors do not become subject to US tax and also so that US non-taxable investors (pensions and universities) do not create inadvetant tax liability for themselves (UBTI in particular)), also simple local compliance i.e. if you are a u.s. citizens and want to buy land in mexico i believe there are certain areas (typically near the coast) where the only way you can do this is to form a mexican entity)
I'm characterizing this "event" as closer to a theft of credit card information than a Snowdon-esque expose of governments. This is a private company who's data was ransacked. Hope the lawyers at that firm are getting ready to be sued by all their private clients.