Why does inductor current lag the applied voltage at its terminal by 90 degrees?

I think the key "physical" reason here is that the induced EMF opposes the change in the supplied voltage. The total voltage across the inductor, that consists of the supplied voltage + the induced EMF, tends to lag behind the supplied voltage as it builds up because of the opposition of the induced EMF. The lag in current is a direct consequence of that.

It is easy to show this using the terminal relation $ v = L di/dt $, but you are looking for another explanation.

It is easier to see that happens to the current when you hook up a voltage source across an inductor if you use the integral form of the terminal relation: $$ i(t) = i(0) + frac{1}{L}intlimits_0^tv(t')dt'. $$

Intuitively, without ever referring to this equation, you know that inductors resist the change in the current through them, in accordance with Faraday's law. Since the magnetic field within the inductor cannot change abruptly with time, neither can the current, which is proportional to the magnetic flux. When you apply time-varying voltage, the current cannot immediately follow this variation, a consequence of which is the phase lag when the voltage is sinusoidal. The mathematical way of saying this is that up to a constant, the current is proportional to the time integral of the voltage.

If you just start applying a constant voltage across the inductor at $t=0$, the current will increase continuously for as long as the voltage is applied. This illustrates the inability of the current to catch up with sudden changes in voltage, and the "delay" in the current response to a voltage input.

In comments you said,

The EMF will be induced only after the current starts flowing.

This is not correct. The EMF in an inductor has nothing to do with the magnitude of the current that is flowing (for example, whether it is zero or non-zero).

It only depends on whether the current is changing.

From a physics point of view, this comes from Faraday's Law of Induction:

$$mathcal{E} = -frac{dPhi}{dt}$$

In the inductor, the magnetic flux $Phi$ is proportional to the current, so we can express this as the constitutive relation of the inductor,

$$V=Lfrac{dI}{dt}$$

You can see it's entirely possible for there to be an EMF produced, even if the inductor current is zero, so long as the rate of change of the current ($frac{dI}{dt}$) is non-zero.

If you connect, at $t=0$, an ideal voltage source to an ideal inductor, the inductor EMF is immediately produced to counter the applied voltage (satisfying Kirchhoff's Voltage Law), and the inductor current immediately starts changing.

The current signal lags the voltage signal if the applied voltage is sinusoidal because the current continues to increase ($frac{dI}{dt}>0$) for as long as the applied voltage is positive, thus it reaches its peak when the voltage just returns to 0 at the end of the positive half-cycle of the voltage waveform.

Another answer has explained that the current lags behind the voltage because the induced EMF in the inductor "cancels out" some or all of the applied voltage.

One way to see why it lags by a phase of 90 degrees, and not some other value, is the fact that unlike resistors, ideal inductors and capacitors do not dissipate any energy. They store the energy in a magnetic field (inductor) or electric field (capacitor) as the applied current (inductor) or voltage (capacitor) increases, and release it again as it decreases. The only phase angles between the current and voltage which don't dissipate any energy over a complete cycle are +90 and -90 degrees. The current lags the voltage by 90 degrees in an inductor, and leads it by 90 degrees in a capacitor.

For real inductors and capacitors, their internal resistance does dissipate some electrical energy as heat, and the phase angle is not exactly 90 degrees. The phase angle in a resistor is of course zero.

Actual back emf is present in the the inductor which opposes the sudden changes in current so there will be time delay of 90 degrees. After that time current increases slowly magnetic field builts up slowly once it reaches maximum there is no opposition it allows full current as it is. Once we disconnected inductor from the supply the supply then it acts source it will supply current in either direction. Every one should get one doubt why we are considering this Current in inductor basically works two ways 1)produce heat (since it's not a pure inductor) 2) produce flux ( main reason for inductors is this) Production flux needs reactive power i.e some amount current which is additionally required and load is not used this current fully it send backs to source or power system. Then why we are considering power factor i.e lagging means 1) basically designer of home appliances are purely active point of they will design that is 90% real power it consumes and 10% reactive power but reactive power must and should there but power generation plants counts the power only for active power only but this 10% reactive power which is not consumed is create many transmission losses because even though 10% huge current will needs for entire city. 2) inductor laggs current by 90 degrees upto 90 degrees back emf present beyond that it won't appears. It opposes current flow upto 90. Similarly at power plants also if pure inductor connected to power plants current supplies after 90 deg there by what happens means additional current load draws for production of flux in inductive loads if current increases voltage decreases so we will increase excitation instead of prime mover torque increasing. So excitation needs some other supply so there by an charges will be applied on consumers and mostly industries, institutions, but not houses domestic appliances