If you would recall, Lenz's law states that the induced current produced in a circuit always flows in such a direction that it opposes the change or cause that produced it. It's similar to inertia. Now, consider a magnet moving towards a conducting loop along the axis of the loop. Since the magnet is moving closer, the magnetic field experienced by the loop is stronger, and so, the flux passing through the loop is increasing. This would cause the induction of a current in the loop that opposes the increase in flux - hence the expression $$-\frac{\mathrm{d}\Phi}{\mathrm{d}t}$$