From left to right:

Assume {A, B ~C} is an inconsistent set of premises. Therefore (A&B&~C) is a contradiction. Therefore, ~(A&B&~C) is true, which by de Morgan's law is equivalent to ~Av~BvC. If ~A, then A, B |-- C is valid vacuously because one of the premises, A, is false. If ~B, the same applies, mutatis mutandis. If C, then the sequent is valid trivially.

From right to left:

Assume A, B |-- C is valid. Assume, for reductio, that A, B, and ~C are all true. Then, since A is true and B is true, by the validity of the sequent, C is true, which means that ~C is false, contra hyp.