This is because the set of possible permutations of 64 bit blocks of plaintext ($2^{64}$ possibilities) to 64 blocks of ciphertext is very high. A key selections just one of these permutations. Even a 256 bit key space is smaller by far than the number of possible permutations.
Some plaintext blocks will likely map to the same ciphertext block for a few of these permutations. But the permutation itself that gets selected by the key is still very likely to be unique, and it is very improbable that you find the matching pairs in the first place.
So your statement "...that means there are many keys which give the same result" is not correct. The result of the key is a specific permutation, and there are plenty of permutations to divide over the keys.
Also see this question as well as this question which explain this for AES/Rijndael specifically.
