Questions of popularity are not going to be easy to nail down (why did everyone suddenly get interested in zombies?) But it's not that actually doing big game hunting was popular ( it was, after all, rather hard for most Victorians to do it) as much as reading about it. That's rather obvious: reading about hunting large dangerous animals that can kill you can be thrilling, the difference between watching a bass fishing competition and seeing Jaws.
As to why some people ( yes, mostly men) in the 19th c. were shooting giant beautiful animals and putting their heads on their walls, it could be summarized best perhaps as an animal-rights activist might summarize it, as a crime: with motive, means, and opportunity.
Motive: the stereotypical big game hunter in a pith helmet in Africa or India was English. Sport hunting had long been a hobby of the landed classes in England .For one thing, they owned the gamelands. Unlike poachers on their lands, they didn't necessarily need to eat what they hunted ( though they did have staff to clean and skin or pluck them) so it was easy for them to get into trophy hunting. Also, hunting the wildlife to keep it away from the crops was thought useful.
Opportunity: When England got an Empire, those upper-class sportsmen were able to simply begin doing the same sport hunting in the new dominions: they were in positions of power and could do so. Also, there was a good bit of game habitat,and the more primitive shorter-range weapons in the hands of the native populations had not reduced the numbers of large dangerous game. So, excitingly large numbers of lions, tigers, cape buffalo, gnus etc. could be found.
Means: though the English wouldn't be limited to spears and arrows, their own gunmakers in the 18th c. had produced rifles and shotguns for small game, birds, and deer. Bringing down rhinos, elephants and buffalo required much more powerful firearms, and by mid-century makers like Westley Richards and Gibbs were producing big game rifles that were at last satisfying the blood-lust of hunters like Sir Samuel Baker, firing belted lead balls weighing as much as two ounces on heavy loads of black powder. These would batter the hunter almost as much as they battered the animals, but as firearms modernized, range and effectiveness did as well. With less punishing rifles to shoot, by the turn of the century, there could be women in pith helmets as well as men, bringing down large animals that they could then pose with for photos.
Although the stereotypical image of big game hunter is of an Englishman in Africa, with a train of native gun-bearers behind him, the willingness of hunters to travel far to find new places to hunt was not limited to the English, of course. Friedrich Gerstäcker traveled to Arkansas territory in the 1840's to simply wander around and hunt- something that was much easier to do on the US frontier than in his native Germany. Americans- notably Teddy Roosevelt- would travel to Africa for big game hunting after 1900. And the thrill of felling large animals was by no means limited to Africa or India: even the historian Francis Parkman would describe killing large numbers of buffalo on his 1847 trip to Oregon, with no apparent reason other than they were simply there and could be shot at.
These two men were also writers, and it is notable how often big game hunters are known chiefly by what they wrote about themselves. Sir Samuel Baker, Richard Selous, Peter Capstick, J.A Hunter, W.D.M. Bell ....all wrote "ripping yarns". With the advances in transportation and firearms technology in the 19th c., in retrospect their success in traveling into new remote habitats and shooting large numbers of large animals seems to be somewhat inevitable and tragic, but it's not until rather late- after 1900- that the tone changes from triumphant, when Selous, Roosevelt and others would begin to express concern about conservation.
