Not so much an “invasion” as there’s only one invader, but the novel Under The Skin by Michel Faber, and the film adaptation starring Scarlett Johansson, are told from the alien’s point of view as she harvests humans for meat. In the novel it’s for an alien conglomerate; in the movie she seems more of a solo traveler.
It’s not actually an invasion story, but it is about weird aliens coming to earth to devour humans – in Larry Niven’s short story Assimilating Our Culture, That’s What They’re Doing (part of his “Draco’s Tavern” series of Bar Tales) the aliens don’t want to go to all the trouble of transporting human bodies across space at vast expense and go through all that butchering and grief, so they surreptitiously obtain copies of our DNA and use the information to clone and grow just the portions they want to eat, hydroponically, using raw materials they already have on their homeworld. So they’ll grow and arm or a leg or, for big blowouts, a whole roasting Torso. Much neater and cleaner and cheaper. They even pay us royalties for the use of it. People agree to this when it’s pointed out that, otherwise, there will still be bootleg clones, and they’d get no money for it, so they might as well profit from the copyright.
Some of the stories mentioned here make me think of a story by Alan Dean Foster, written in 1971 called “With Friends Like These…” It was included in an anthology of the same name a few years later.
A group of aliens come to earth in desperation as their federation of worlds is under attack by a powerful invading species. In the distant past the earthlings were a powerful race that was spreading out too fast so the aliens pushed them back and placed force field around earth to contain the violent earthlings. The aliens are disappointed as first as it seems that the earthlings have devolved to a simple farming planet. Later it is revealed that the simple life is a ruse and that the humans have developed to an insane and powerful degree. They agree to effortlessly destroy the invaders but now the delegation wonders what will happen to them.
In the story, aliens put up a Shield around Earth millennia ago to stop a war between humans and another alien species. Now another group of aliens has come to lift the Shield because they’re losing a war and are desperate for allies, and humans are the legendary boogeymen of the galaxy.
At first it seems to the visiting aliens that humans have reverted to a peaceful agrarian society, but they soon discover human technology is incredibly advanced, they’ve been waiting for someone to turn off the Shield (it couldn’t be done from the inside) and the whole planet is a spaceship.
P.S. I just re-read the story for the first time in decades, which is available for checkout here:
In the forward and afterward, Alan Dean Foster commented that it was inspired by Eric Frank Russell as a Terra über alles story after a conversation with John Campbell (long-running editor of Astounding/Analog magazine), and that it was actually his first published story.
I don’t recall that exact James White short story, because of the plot details you give, but “Red Alert” is about aliens secretly arriving on Earth to tamper with human societies by manipulating media and (telepathically) the minds of the nations’ leaders, ratcheting tensions and getting nations organized for World War III. The implication is that the aliens will provoke humanity into nearly extincting itself in the Final War and then mop up the survivors. But, with the twist you describe: the organization and mobilization is to get the world so busy being ready to fight each other and so frightened of the pending war, that they are actually relieved when the world’s leaders are telepathically told the truth: there will be no war. The alien fleet is there to rescue Humanity from the coming nova explosion of the Sun, an unnatural stellar explosion caused by a navigation accident of a ship of the interstellar culture of the “invaders”. The war scare helped keep the aliens hidden and the mobilization made it easy to marshal humans into an orderly evacuation.
Your mention of Eric Fran k Russell reminds me of one of his novels – Wasp. It’s not an invasion story, but it tells of the adventures of a (human) saboteur disguised as an alien on their homeworld performing minor acts of sabotage to screw up production of war materiel and destroy morale.
It actually could almost have been a non-sf story about subversive tactics in wartime. An interesting read.
I recall another sneaky humans story. It starts just as we lose one interstellar war. The aliens take a lot of our War Leaders as hostages, who will be kept in suspended animation for a long time (like, a century, maybe?) in order to keep us in line.
Their secret is, they expect the humans to be so humiliated by the loss that our culture will turn to hedonism and nihilism, sapping our strength and will to fight. The bulk of the story is the aliens studying us for the next century, tracking the rise in drug use and participation in non-military pursuits. As all the statistics show, ther plan is working, and humanity as a whole is sinking into despair.
But, at the last minute, as they are about to return our War Heroes to us, one of them asks, “Hey, what would we have done if we realized Our Great War Hero was about to come back after a century, and judge our modern degenerate society?” Turns out, the whole human Race has been faking it all along. We pretend to take drugs, and all the aliens do is look at the charts that report every-increasing drug use. They never do the dirty work of confirming that we’re actually using those drugs.
But it’s too late, the War Heroes are already disembarking, to be met by those non-military members of the “Sporting parachuting and target shooting society”, that is, the elite Airborne Regiments of Earth.
That’s The Underhandler, another one by Christopher Anvil.
Another Asimov: What is this thing called Love? (originally Playboy and the Slime Gods) is about alien researchers who are investigating whether the sexual reproduction methods of humans are dangerous and should be curtailed. (The aliens reproduce asexually.)
I’m sure that I read this one in The Illustrated Man, by Bradbury. Full scale Martian invasion has succeeded, but with an effect similar to what Pratchett would later describe as the experience of invaders of Ankh-Morpork. It’s told from the perspective of one of the Martians, a conscript in the invasion force who doesn’t predict the Ankh-Morpork angle precisely, but his reading of tons of Earth’s pulp SF has convinced him that the invasion will end disastrously for the Martians.
@Andy_L would probably be able to name the story.
There’s an old comic book where we see the aliens discussing their heh-heh-heh plans to send a disguised operative to walk among us, having crafted a face and clothes and et cetera for the guy based on the TV broadcasts they’ve painstakingly monitored: the clothes are perfect, the face is perfect — honestly, they didn’t miss a single detail in any of the shows they watched in glorious black-and-white.
Although the aliens aren’t stupid - they do come up with plans that use humanity’s cleverness against us, successfully, as I recall
“Those ships will be the biggest damn cosmic rays the universe has ever seen!”
The Concrete Mixer by Bradbury in the Illustrated Man, as you suggest
In the story, a warlike race of Martians plans their glorious conquest of Earth but one of them, Ettil Vrye, foresaw defeat. He was given his choice of joining the Legion of War —or burning!
This thread on Scifistackexchange mentions the SDMB
Home, featuring the voice of Jim Parsons
The story follows the shared adventures of a friendly alien who is shunned by the rest of his kind, and a teenage girl searching for her mother after they are separated during an invasion of Earth.
This is based on the fabulous kid’s novel The True Meaning of Smek Day, whose author is a real gem of a humorist.
One interesting bit of coincidental trivia: The story takes place in a future where the former star baseball pitcher Fernando Valenzuela did die of natural causes, as some human characters were commenting before the aliens appeared!
As Fernando passed away just recently in our universe…:
Keep watching the skies… now!!! 
You forgot to mention how he started having all of his middle-age/elderly male protagonists leave their wives and shack up with attractive younger women like he himself had done.
Also it ends on a funny note when the somewhat condescending human advisers are told that oh by the way, the empire has just discovered a new race that appears to be about as much smarter than humans as humans are to the imperials; so, what do they suggest?