At t=0 the voltage across the inductor equalizes the battery voltage because by the time instant you start running a current through the inductor, a back EMF of the same value as the battery's voltage is induced, however this induced EMF, according to Lenz' Law, will always oppose the polarity of the battery. Meaning, the inductor at t=0 acts like another battery of the same voltage of opposite polarity, hence the voltage across the inductor being equivalent to battery voltage.

Since there is zero current then what makes the inductor voltage equal to battery voltage?

Well that's the point! The reason why there's zero current is because the inductor's voltage is equal to battery voltage even when a series resistor is applied. Which is an equivalent to an ideal open circuit.

The reason why voltage across the inductor falls overtime is because current cannot stay zero. If it did, the changing magnetic field wouldn't be there, therefore no back EMF to restrain the current from running through the inductor. This means that the current will increase overtime -- meaning the inductor will becomes less of an open circuit. Basically the inductor will act like a resistor whose resistance value drops over time. This means that voltage across the inductor drops, until you reach the point where that interpretated resistance drops to zero and your inductor acts as a dead short.

At t=0 the voltage across the inductor equalizes the battery voltage because by the time instant you start running a current through the inductor, a back EMF of the same value as the battery's voltage is induced, however this induced EMF, according to Lenz' Law, will always oppose the polarity of the battery. Meaning, the inductor at t=0 acts like another battery of the same voltage connected in reverse, hence the voltage across the inductor being equivalent to battery voltage.

Since there is zero current then what makes the inductor voltage equal to battery voltage?

Well that's the point! The reason why there's zero current is because the inductor's voltage is equal to battery voltage even when a series resistor is applied. Which is an equivalent to an ideal open circuit.

The reason why voltage across the inductor falls overtime is because current cannot stay zero. If it did, the changing magnetic field wouldn't be there, therefore no back EMF to restrain the current from running through the inductor. This means that the current will increase overtime -- meaning the inductor will becomes less of an open circuit. Basically the inductor will act like a resistor whose resistance value drops over time. This means that voltage across the inductor drops, until you reach the point where that interpreted resistance drops to zero and your inductor becomes a dead short.