Much like the more recent "Allan Houston Rule," the "Larry Bird Exception" was not actually used on Larry Bird. When the NBA instituted its salary cap, it allowed teams to exceed the cap in order to re-sign their own player. The intention of the rule was to make sure that teams could retain their stars and ensure continuity.
When the cap was agreed to in the 1983 collective bargaining agreement, Bird was slated to be a free agent after the ’83-’84 season. The exception was developed so that the Celtics could keep Bird in Boston and not lose out to a higher bidder.
The new CBA allowed teams to peg their cap to whatever they spent in the 1983-84 season, so in the summer of 1983 it behooved teams to spend lavishly. With this in mind the Celtics re-signed Bird to a seven-year, $12.6 million contract, preventing him from becoming a free agent the next year, and thusly preventing themselves from having to use the Bird exception on Bird himself the next offseason. Once the cap kicked in during the 1984 offseason, the Celtics did use their Bird exception, but they used it on Cedric Maxwell.
The Bird exception still exists, allowing teams to exceed the salary cap to sign qualifying veteran free agents (those who have played at least three consecutive seasons with one team) for up to the maximum salary.