I know it has something to do with 2's complement and adding 1, but I don't really get how you can encode one more number with the same amount of bits when it comes to negative numbers.
Think about it in these terms. Take a 2-bit number with a preceding sign:
000 = 0 001 = 1 010 = 2 011 = 3
Now let's have some negatives:
111 = -1 110 = -2 101 = -3
Wait, we also have
It has to be negative, because the sign-bit is 1. So, logically, it must be -4.
(Edit: As WorldEngineer rightly points out, not all numbering systems work this way -- but the ones you're asking about do.)
Because there are not two classes of numbers in the integer range, but three: negative numbers, zero, and positive numbers. Zero has to take up a slot (would be rather impractical not to be able to represent zero...), so either the positive or the negative class has to give up a slot. The fact that it's usually the positive range that has to make that sacrifice is to a certain extent arbitrary, but on the level of bit manipulations there are some things that this decision makes more convenient.
There are BASICALLY three ways to represent signed integers in binary: 2's complement, 1's complement, and sign-magnitude. (Biquinary went the way of the Dodo Bird a long time ago.)
1's complement and sign-magnitude have two zero values, +0 and -0, each with a unique representation. 2's complement only has one zero value, and one representation.
Now, a field of N bits can encode 2^N values. Subtract one in 2's complement, and you have 2^N-1 = 2^(N-1) + 2^(N-1) + 1. Since the representation for zero is all zero bits, and a + sign is zero, there will be one more possible nonzero representation with the sign bit set to 1.
This is a very long-winded way of saying 2's complement represents values in the range -(2^(N-1)) .. +(2^(N-1) - 1).
1's complement actually has an advantage over 2's complement if you are doing integer digital signal processing computations. 1's complement operations inherently truncate toward zero. 2's complement truncates toward -infinity. I learned this one the HARD way...