So surprisingly, a lot of people are debating what caused the drought in the original film. People say that it wasn't Scar's fault and that he was a good king, or that it was Mufasa's ghost's doing. I'm going to explain how it WAS Scar's fault and how the hyenas played a big role in it.

The induction of the hyenas into the pride would've caused a massive change in population. Lions and hyenas are apex predators. Eaten by no one and feared by all. If they were in the same territory, they'd have to fight for enough food to survive. That's most likely why Mufasa banished them to the Elephant Graveyard (the dialogue between Shenzi and Banzai implies that they at one point tried to take over the Pride Lands, but were cast out by the lions). Hyenas are also competitors with not only lions but other predators as well, such as cheetahs, leopards, wild dogs, and caracal cats.

With more predators, there would be less and less prey around.

But why would there be little plants? Wouldn't the vegetation rise?

This is normally what gets people to think the drought wasn't Scar's fault, but here is something which people don't know about: Keystone species.

A keystone species is so important to the environment, that the habitat would be nothing without it. So essentially, the plants would die without it. 

Lion King takes place in Tanzania, Africa. A large keystone species is the elephant. They eat the acacia plants that thrive off of trees, and essentially knock them down, which keeps the savannah from becoming a forest, and it also helps more brush and grasses grow, which help the herbivores such as zebras and gazelle.

So if we added too many predators, in no time, the keystone species would be taken out, which would not only affect the plants it eats but the plants' other predators eat.

The induction of the hyenas would strip the land bare of its resources, including the keystone ones, which proved to be a detriment to the plants as well. And this would all be during a drought, in which there are already too few plants. With this, the surviving herds would be forced to move on, both from the lack of food source due to the loss of the keystone species and from the multiplied predation.

In addition, we learn from the sequel that there were lionesses who supported Scar's reign despite how detrimental it was. If they supported Scar, who didn't care what he was doing, they would likely have not had a regard for the Circle of Life as well and would've been over-hunting just as much as the hyenas were, further disrupting the Circle.

In semi-canon media, it's implied that the hyenas weren't the only cause of the famine, but instead, the lionesses were forced to provide food for themselves, and the hyenas. Now, I say semi-canon, because the only 2 places where this information was given was in a deleted scene, and in the musical, but they weren't in the finalized original film, so they can be argued as not fully canonical. However, this would also prove to be terrible for the environment, since once again, removal of the herbivores would screw over the plants, bite the other herbivores in the chassis, and starve the predators.

Simba returning to the Pride Lands wasn't a "magical" return. At the end of the film, we see a transition from Simba's coronation to Kiara's presentation, which would be a gap of 3 months. Within those 3 months, both Zira's followers, and the hyenas would've been exiled from the Pride Lands. With the over-consumers gone, and the drought coming to an end, there would be less predation, allowing the herds to return, including the keystone species, and that would help the plants come back, and ultimately, the Circle of Life would re-build.

In SP, we see fewer animals during Kiara's presentation, which makes sense since it was only 3-4 months since Scar was killed, and the land was slowly re-building, and we see a section of the Pride Lands that still suffered from Scar's reign.

So I don't think there was any magical lion god or a long, LONG drought that was the reason for the famine, but it was rather over predation and over-population during a feeble drought that collapsed the kingdom.