Imagine we could get away with a one-dimensional treatment and forget about all the complications of three-dimensional orbitals. We could then represent the wavefunction of the hydrogen atoms by Gaussians around the nuclei. For simplicity, we fixed them at a distance of two units.
Obviously the wavefunction is symmetric with respect to exchange of the positions and there is a probability to find both electrons exactly in the middle between the atoms. We associate this function with a bonding state of reduced energy.
Likewise, we can define an anti-symmetric wavefunction where the probability of finding an electron exactly between the atoms is zero. This would probably make the atoms fly apart and is often called an "anti-bonding" state.
The figure below plots the modulus of the two orbitals; in reality the atoms should be closer together, such that the wavefunction of the bonding state assumes a single maximum exactly between the atoms. This is not the case here, but just assume that we hold the atoms on a stick at this distance for illustration. The difference between the bonding and the anti-bonding states is still visible because the probabiltiy of finding an electron between the atoms is finite for the bonding state whereas it is zero for the anti-bonding state.
In molecular-orbital theory the states are normally ordered with respect to their energy. Starting from the energy level of the isolated atom, there is the bonding state at a lower energy and the anti-bonding state at a higher energy. The colored blobs to the right can serve as illustration which signs we actually used to construct our wavefunctions; positive and negative signs are denoted by red and blue, respectively.
The momentum-representation is totally equivalent, but it is maybe a little less intuitive. Essentially the figures show the probability to find an electron with a certain momentum. Clearly, the most probable momentum of the bonding state is zero. For the anti-bonding state, zero-momentum is actually excluded (maybe a sign that this state would fly apart immediately if we hadn't pierced the atoms on a stick).
The next section will make clearer why we want to use the momentum-representation.