It comes down to how dividing by two partitions the naturals. You can always get from 0 to n questions right, and you pass if you get more than 50% of them right. For the odd numbers, this means that you have to get n/2+1 or more questions right, where n/2 is integer division: so, for example, if the test has five questions, you have to get at least 5/2+1 = 2+1 = 3 right. That's like that because, as integer division throws away the rest, n/2 falls short of the 50% you need to pass, so you have to get one more question right. This means that you fail if you get from 0 to n/2 right, and you pass if you get from n/2+1 to n right. You'll notice that there are exactly n/2 possibilities for each: so, for example, in the 5 example you fail with 0,1, or 2 and you pass with 3, 4, or 5. The probabilities for each possiblity are not the same, but thanks to symmetry(getting exactly 4 questions right is the same as getting exactly 1 wrong, so it's the same probability as getting exactly 1 right) that doesn't matter: getting 0 right has the same probability as getting 5 right and the same for all the others, so both chunks have exactly the same probability. This gives you an 1/2 chance to pass.

For the even numbers, getting n/2 questions right is enough, since that is already 50%. That is, you pass if you get from n/2 to n questions right, so you only fail if you get from 0 to n/2-1 right. So now you have n/2 ways of failing and n/2+1 ways of passing! For example, with n=6, you fail with 0, 1, 2 and you pass with 3, 4, 5, 6. Symmetry tells us again that the probability of getting from 0 to n/2-1 right is the same as getting from n/2+1 to n right, but this leaves the middle possibility, 3, out! That's why passing is slightly more likely now.