These are the remains of Roman London's Amphitheatre, discovered during excavations beneath the Guildhall in 1988, and included in the new art gallery when it reopened in 1999 (having been closed since it was bombed out in 1941).
To be honest the whole experience is rather underwhelming, but nevertheless it is part of London's history and it is a reminder of how our city has changed.
It's open 10am-5pm Mon-Sat and 12-4pm on Sundays. Entrance is included in the price of the art gallery at a very acceptable £2.50, and it's free if you can prove you live or work in the City of London. For more information click here.