A combinatorial proof of a more general inequality is given by Douglas Woodall.
One line proof Fisher's inequality is given by Renaud Palisse
Palisse, Renaud, A short proof of Fisher's inequality, Discrete Math. 111, No.1-3, 421-422 (1993). ZBL0788.04003.
Woodall, Douglas R., A note on Fisher's inequality, J. Comb. Theory, Ser. A 77, No.1, 171-176, TA962729 (1997). ZBL0878.05011.