The title of Wikipedia's page "Kingsway tramway subway" has way too many 'way's, but the page is very informative.
The tunnel was constructed 1906, enabled by the 'slum clearance' project in the Aldwych / Kingsway area. There were two stations in the tunnel: at Holborn and Aldwych. Trams ceased using the tunnel in 1952, after which it was mainly used for storage. It was opened for north-bound motor traffic in 1964 in an effort to reduce congestion around the Aldwych.
You can see two northern entrances/exits to the tunnel, one in Southampton Row, just south of the junction with Theobald's Road, and another in Kingsway just north of the Aldwych. The northern one was used by the trams. The southern one was constructed in the 1960s as an exit for motor traffic.
The southern entrance/exit has changed twice. Originally the trams entered/exited from the Embankment, just to the west of Waterloo Bridge (shown in this photo). When the bridge was rebuilt during WW2 that entrance/exit was moved to be centrally below the bridge. In the 1960s that was closed off (and the space used to accommodate a business) and a new entrance created for traffic from the bridge itself.
Stories of London has many images that help to make sense of this history in words. The image on this page shows a tram exiting the tunnel when the mouth was on the Embankment to the west of the old Waterloo Bridge.
