The voltage is not negative, always. The negative sign in Faraday's law (Lenz's law) does not mean that the EMF (or current) always points in some "negative" direction. It means that the current always flows in a way to oppose the change in flux, which is nicely illustrated in that video clip.

What happens when emf is negative?

Negative EMF generates a current that creates a magnetic field which opposes the change in magnetic flux.

Why emf is sometimes positive and sometimes negative?

Say there is a sheet and magnetic field is going through it from front to the back, we can call the flux there as positive and negative when it's the other way round. Basically, the magnetic field produced due to the induced current opposes the magnetic flux producing the current itself.