Suppose f:G1->G2 is a graph isomorphism, and G1 is bipartite. We need to show that G2 is bipartite.
Let V1 and V2 be the vertex sets of G1 and G2. G1 is bipartite means we can partition vertex set V1 into two disjoint sets X and Y, such that V1=XUY, and such that every edge of G1 connects a vertex x of X with a vertex y of Y.
Now consider the subsets f(X) and f(Y) of V2. Since f is a bijection from V1 to V2, f(X) and f(Y) are disjoint and f(X)Uf(Y)=V2.
Now consider two vertices w,z of V2. If w and z both lie in f(X), then f-1(w) and f-1(z) both lie in X. Since G1 is bipartite, there is no edge between f-1(w) and f-1(z), since f preserves adjacencies, there is no edge between w and z in G2. Similarly, if w and z both lie in f(Y), there is no edge between them.
Every edge of G2 connects a vertex of f(X) with a vertex of f(Y), so G2 is bipartite, with bipartition (f(X),f(Y)).