Selecting an allied nation's Library will reveal their current research progress. Getting information from an enemy nation's Library requires a Spy to plant an informer.
Future technology is extremely useful, but takes a long time to finish researching unless you have the Super Collider wonder. While researching future technologies, your Libraries are destined to become your enemy's favorite target, so building additional Libraries may help deter the disgusting surprise of a future technology stopping when it's about to complete because the enemy blew up your only Library. As long as a single of your Libraries still stand, research won't stall.
Additional Libraries also allow for simultaneous research of regular technologies.