A proof of the statement has already been given, so I will just add a small historical remark. The left hand side of your identity is the Veneziano amplitude in the case of four identical scalar particles. The right hand side corresponds to another crossing symmetric four-point amplitude for scalar particles suggested by Virasoro in "Alternative Constructions of Crossing-Symmetric Amplitudes with Regge Behavior". These are known to coincide for the choice of parameters $a+b+c=1$ giving your identity. I would recommend the article "The birth of string theory" for a quick survey of the Veneziano model and its generalization to N scalar particles and related mathematics.

A proof of the statement has already been given, so I will just add a small historical remark. The left hand side of your identity is the Veneziano amplitude in the case of four identical scalar particles. The right hand side corresponds to another crossing symmetric four-point amplitude for scalar particles suggested by Virasoro in "Alternative Constructions of Crossing-Symmetric Amplitudes with Regge Behavior". These are known to coincide for the choice of parameters $a+b+c=1$ giving your identity. I would recommend the article "The birth of string theory" for a quick survey of the Veneziano model and its generalization to N scalar particles and related mathematics.