Reviewing an omnibus is a tricky venture - this volume contains 5 books and one essay, covering a number of different topics. Thanks to the inclusion of that final essay, we can't even say that there is a unifying Anglo-Indian element to the book. If we fudge a little by excluding that essay, however, we can say that this is the story of the wilds of India during the last days of the Raj, and a testament to the courage & ability of her people.
Man-Eaters of Kumaon was my introduction to Corbett, and I've read it 3 or 4 times over the last few years. It's a collection of true stories (as are all Corbett's writings) about his work hunting man-eating tigers. It's filled with extremely vivid depictions of jungles and tigers, and a sense of deep love for India and the Indian people. If you're only going to read one Corbett book, make it this one.
The next two books in the omnibus are The Temple Tiger & The Man-eating Leopard of Rudraprayag. They're both more of the same - if you really liked Man-Eaters, then you'll want to read these, but if you didn't, then I can't say they offer anything new. I will mention that there was one page in The Man-Eating Leopard that offered a laugh out loud moment immediately followed by one of the most bloodcurdlingly terrifying things I've ever read, which was itself immediately followed by comic relief. It's the best kind of roller coaster, and perhaps worth reading for that one page alone.
My India is rather surprising change of pace - it has little to no hunting in it whatsoever. Dedicated to "my friends, the poor of India", it's a collection of incidents & biographical sketches of various Indians that Corbett knew, notable not because of their high rank or wealth, but because of their strength of character or their fortitude under impossibly trying circumstances.
Jungle Lore is another change - though not quite a how-to guide, it's an extended meditation on living in the jungle, and learning how to read it & listen to it. This is perhaps the weakest entry in the collection, but still worth reading if you've enjoyed the rest of it.
There are, of course, two issues that may give people pause when contemplating reading Corbett. Hunting tigers, an endangered species, is definitely less acceptable now than it was in his day, and some people might not want to read about it. If that presents a problem for you, I can more or less put your worries to rest.
There's a common misapprehension that Corbett exclusively hunted man-eaters, but that isn't entirely true. It was only in his final decades as a hunter that Corbett refused to kill a big cat unless it was actively hunting people for food. Before that, his primary focus was on man-eaters, but he would also sometimes kill a tiger or leopard if it was preying on the herds of poor farmers - he loved big cats, but he loved poor Indians more.
Before that stage of his life, it does seem that he hunted for sport, and helped others to do the same - in fact, an incident in Jungle Lore makes it clear that even during the part of his life where he himself only hunted man-eaters or nuisances, he would sometimes help others hunt tigers for sport. Nevertheless, I'd say that 90% of his stories are indeed about tigers that absolutely had to be killed - you can grit your teeth and appreciate the rest for his prose (or even skip them, if necessary).
The other big hurdle is race - as a white Angl0-Indian surrounded by & employing poor Indians, you'd expect a whole lot of racism. Surprisingly, there's almost none present. There's a little paternalism, perhaps, but it's all in the least toxic form possible - building walls to protect villages from tigers, convincing the government to issue villages with first aid kits, and so on. When individual people are singled out in a story, it's all done with immense respect. The only line in the entire collection that even approaches racism is when he says that Indians from a certain province are all awesome people who make for great friends.
In short, The Jim Corbett Omnibus is a fine addition to any library. I highly recommend it.