"prominent, important, first-rate," by 1925, a figurative use from baseball, where big league was used for "a major league" by 1891. See league (n.1).
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league n.1
"alliance," mid-15c., ligg, from French ligue "confederacy, league" (15c.), from Italian lega, from legare "to tie, to bind," from Latin ligare "to bind" (from PIE root *leig- "to tie, bind"). Originally among nations, subsequently extended to political associations (1846) and sports associations (1879). League of Nations is attested from 1917 (created 1919).