Thought experiment
Let us have a thought experiment. At first, we fixate some configuration for atom or molecule with N electrons, choose two electrons with same spins and give them indices “1” and “2”. Let the wave function for this configuration have some value Ψ0 and Ψ0 > 0 for definiteness (the sign of Ψ0 does not matter). Then the wave function must change sign and turn to –Ψ0 when we change places of electrons indices “1” and “2”. We take with fingers or tweezers electrons “1” and “2” and move them in some arbitrary way changing their places. The other electrons and atom nuclei stay still. In this situation electron “1” runs some trajectory and electron “2” runs another trajectory in physical space. The two trajectories in three-dimensional physical space can be described by one line in six-dimensional configuration space named by us the interchange trajectory. The spins can be ignored because they coincide.
It turns out that any choice of the interchange trajectory gives Ψ=0 at some moment. (For example, we can move electron “1” far away in atomic scale, electron “2” put on its place and then choose some intricate way to move electron “1” to the former place of electron “2” to avoid symmetrical situations). It follows from continuity of the wave functions and the well-known mathematical analysis theorem that proves a continuous function determined on some segment and having opposite signs on its ends turn to zero at least at one point of the segment. We consider the interchange trajectory as such segment. There can be more than one zero points but their number will be odd necessarily. The nodal surfaces of the wave function in configurational space consist of these points.