When you apply a DC voltage source across a perfect inductor current ramps up at a rate equal to V/L. This ties in with the equation V = Ldi/dt.
However, there is a controlling mechanism inside an inductor that prevents the current rise being infinite straight after voltage is applied. This is usually referred to as a back emf. That back emf opposes the applied voltage and is therefore negative. This gives rise to the equation V = -Ldi/dt.
You can visualize this back emf if instead of a single coil inductor you had two perfectly coupled windings sharing a common ground. When you apply (say) +10 volts to the driven winding, +10 volts appears on the terminal of the "secondary" and it is +10 volts with respect to ground so, if you accept that the same +10 volts is also induced in the driven winding then it has to oppose the driving voltage.
That voltage is produced by the changing flux of the forward current in the driven coil rising positively. There is no reason to suppose that the driven coil doesn't also have this opposing voltage because it is subject to the same changing flux. That bit confuses engineers a lot but it is true for electrical engineering.
Now I'm potentially in deeper water because, you could then argue that if it is exactly +10 volts then surely no current can flow in the first place. The bottom line is (without going in too deep) is that the back emf and forward voltage are not quite the same. This then becomes a physics question!