Pi bond need sigma bond to exist
I think it would be more correct to say that pi bonds tend to exist in conjunction with sigma bonds, rather than that they need sigma bonds to exist - as pointed out in the comments. 

This can be explained rather easily by basic molecular orbital (MO) theory, which seems like an appropriate framework for the question. Let's consider the simple case of a homoatomic bond such as C=C. When you have two C atoms coming into contact so as to form MOs, you form sigma MOs (bonding and antibonding) from the set of atomic S orbitals, and pi MOs (again, bonding and antibonding) from the Px and Py atomic orbitals. 

Among the (molecular) orbitals, the sigma bonding MO generally has the lowest energy, followed by the pi bonding MOs. This means that the sigma bonding orbitals will 'fill' before the pi bonding orbitals. Because electrons preferentially occupy MOs from lower to higher energy, as long as the sigma bonding MOs are lower energy than the pi bonding MOs you will see sigma bonding orbitals populated before pi bonding MOs. Thus, to have a pi bond (occupied pi bonding MOs), you generally also have a sigma bond (occupied sigma bonding MO). 

As pointed out in the comments to your question, this is somewhat of a generalization - but I think it's an accurate description of the phenomenon your instructor described.