The primary advantage of the design is that the wing's leading edge remains behind the shock wave generated by the nose of the aircraft when flying at supersonic speeds, which was a distinct improvement on traditional wing designs. Another advantage is that as the angle of attack increases the leading edge of the wing generates a huge vortex which remains attached to the upper surface of the wing, making the delta have very high stall points. The combination of these two features is a dream come true, a normal wing built for high speed use is typically dangerous at low speeds, but in this regime the delta transitions to a mode of lift based on the vortex it generates.
Lippisch studied a number of ramjet powered (sometimes coal-fueled!) delta-wing interceptor aircraft during the war, one progressing as far as a glider prototype. After the war Lippisch was taken to the US, where he ended up working at Convair. Here the other engineers became very interested in his interceptor designs, and started work on a larger version known as the F-92. This project was eventually cancelled as impractical, but a prototype flying testbed was almost complete by that point, and was later flown widely as the XF-92. The design generated intense interest around the world. Soon almost every aircraft design, notably interceptors, were designed around a delta-wing. Examples include the Convair B-58 Hustler, the Avro Arrow and the MiG-21.
Deltas fell out of favour due to some undesirable characteristics, notably flow-separation at high angles of attack (swept-wings have similar problems), and high drag at low altitudes. This limited them primarily to the high-speed, high-altitude interceptor roles. A modification, the compound delta, added another much more highly swept delta wing in front of the main one, to create the vortex in a more controlled fashion and thereby reduce the low-speed drag.
As the performance of jet engines grew, fighers with more traditional planforms found they could perform almost as well as the deltas, but do so while maneuvering much harder and at a wider range of altitudes. Today a remnant of the compound delta can be found on most fighter aircraft, in the form of leading edge extensions. These are effectively very small delta wings placed so they remain out of the airflow in cruising flight, but start to generate a vortex at high angles of attack. The vortex is then captured on the top of the wing to provide additional lift, thereby combining the delta's high-alpha "trick" with a conventional highly efficient wing planform.