"Birdland" is a song paying homage to one of the most influential jazz saxophonist of the 20th Century: Charlie Parker. Parker obtained the nickname "Yardbird" which was eventually shortened to "Bird". When an aficionado of bebop jazz spoke of Bird, everyone knew who he was talking about.
The song made popular by The Manhattan Transfer is replete with references to famous jazz artists that played with Parker at the famous venue where they played.
"Bird would cook, Max would look" obviously refers to Charlie "Bird" Parker and Max Roach, a stylistic drummer of the bebop period.
"Miles came through", refers to the great Miles Davis, considered the premier jazz trumpeter of the post-WWII era. "'Trane came too" is in reference to John Coltrane, a key tenor saxophonists who often played with Miles Davis and Thelonious Monk.
"Basie blew, Blakey too" refers to Count Basie, a jazz pianist and band leader from Charlie Parker's home town of Kansas City and Art Blakey, another influential bebop drummer known for his hard driving drumming technique.
"Cannonball played that hall", is Julian "Cannonball" Adderley, a jazz alto saxophonist who was a member of the Miles Davis Sextet and later headed his own quintet.