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(a) Let's assume A is lying. Then no one among A and B are knaves, so they have to be knights, but knights never lie. We came up with contradiction, so A is telling truth. A is a knight. Since one of them is a knave, the knave must be B. (A - knight, B - knave)
(b) Let's assume A is knight, then B is a knight too. Since that is just what A said, and if he is a knight, his statement must be true. But look what we have here: we just showed that if he is a knight, then so is B.This is exactly what he claimed, and we’ve just seen this statement is true. Since he said a true statement, he must be a knight, and so therefore B must be too. (A - knight, B -knight)
(c) Assume A is a knight, then A and B should be knights. B as a knight must tell truth, but it is not consistent with what B said. Let's assume A is a knave, then there B can be either a knight or a knave. If B is telling truth, then he is a knight and they are indeed knave and knight as B claims. If B is lying, then negation of "either A is a knight, or I am a knight" gives "A is not a knight and B is not a knight", that is consistent with what B claims. (A - knave, B - knight or knave)