The narrator and other characters make several connections between people and animals. And we're not talking about little references here and there that are hidden in symbols or whatnot – there are BIG paragraphs in which people are compared to animals in overt ways. For instance, when the narrator comes out of the ruined house and sees the world, now totally changed by the Martians, he feels something new:
For that moment I touched an emotion beyond the common range of men, yet one that the poor brutes we dominate know only too well. I felt as a rabbit might feel returning to his burrow and suddenly confronted by the work of a dozen busy [workers] digging the foundations of a house. I felt the first inkling of a thing that presently grew quite clear in my mind, that oppressed me for many days, a sense of dethronement, a persuasion that I was no longer a master, but an animal among the animals, under the Martian heel. With us it would be as with them, to lurk and watch, to run and hide; the fear and empire of man had passed away. (2.6.2)
You can see this isn't an abstract thought. The narrator doesn't just say, "Well, I guess we don't have power anymore." He's saying "We don't have power anymore. We're just like animals." And we're sure you can imagine other comparisons he could make in order to say that: "Oh, we don't have power anymore. We're just like plants or children." But the narrator doesn't make those comparisons. There's something important about the human-animal comparison.
Wells chooses to compare people to animals because, when we're faced with the Martians, we are as powerless and confused as animals would be. After all, does the rabbit understand that people are building a house? No way. The rabbit just probably screams some rabbity curses, like "What the fruit!" before getting out of there.
Since there are a lot of comparisons made to animals throughout the book, how does that affect our reading process? Well, for one thing, we think that the animal comparisons are easy to imagine. A lot of people have some direct experience with animals that demonstrates how they are weaker or just plain afraid of us. For instance, many Londoners would have the experience of walking through a city and having pigeons scatter. And that's just what the people do when a Martian shows up.
Not only is it easy to imagine, but Wells may be asking us to rethink our relation to animals. As the narrator notes, "Surely, if we have learned nothing else, this war has taught us pity – pity for those witless souls that suffer our dominion" (2.7.3). Let's expand this idea. At the end of the book, the narrator remarks that people now have a better idea of how there's "a commonweal of mankind" (that is, we're all in this together – what's the point of England and Germany fighting for colonies in Africa if the Martians are just going to come in and kill us all?). But what if the idea at the end is even bigger than that – not just a commonweal of mankind, but a commonweal of the Earth?
In fact, if we look back at the opening of the book, we find that the narrator points out that humans have done some bad caretaking of the world. Not only have we killed off other humans (like the Tasmanians), but we've also killed off the dodo and the bison (1.1.6). Maybe the human-animal comparisons are meant to teach us that we're all in this together as Earthlings. And, if we're so closely connected, then maybe we should be a little more careful with what we (humans) do to them (animals).
Historical Side Note: In 1896, Wells wrote The Island of Dr. Moreau, which is all about the connections between animals and people. The idea that people needed to be more responsible in relation to nature was growing throughout the 19th century. Don't believe us? Check out this timeline:
1824: The Royal Society for the Prevention of Cruelty to Animals was formed in Britain
1866: The American Society for the Prevention of Cruelty to Animals was formed
1898: The British Union for the Abolition of Vivisection was founded, which was a group that argued against animal experiments
1892: The Sierra Club was established
Around 1900: Theodore Roosevelt did a lot of work setting aside land to use as American national parks
So, yes, the idea that people need to be more thoughtful about animals was getting more attention around the time when Wells wrote The War of the Worlds.