On 28 June 1914, Archduke Franz Ferdinand and his wife Sophie were shot dead in their car by 18-year-old Gavrilo Princip directly outside this building. Inside is a one-room exhibition, part of the Museum of Sarajevo, which gives a cursory examination of the city's Austro-Hungarian–era history and the assassination which would ultimately be the fuse that detonated WWI.
There's not a lot to see here; even the assassin's gun displayed is a replica, the real one being in a Vienna museum. You may be content to skip the entry fee and instead examine the sepia photos conveniently positioned in the front windows which effectively illustrate that momentous day's events.