"Legwork", a novelette by Eric Frank Russell, which was also the answer to the question Alien scout outsmarted by humans; first published in Astounding Science Fiction, April 1956, available at the Internet Archive and the Luminist Archives. You may have read it in one of these compilations.
A lone alien landed on Earth.
As nearly as an Andromedan thought form can be expressed in print, his name was Harasha Vanash. The formidable thing about him was his conceit. It was redoubtable because justified. His natural power had been tested on fifty hostile worlds and found invincible.
[. . . .]
So it was with nonchalance that he landed on Earth. The previous day he'd given the planet a look-over and his snooping had set off the usual rumors about flying saucers despite that his ship resembled no such object.
He arrived unseen in the hills, got out, sent the ship up to where its automechanisms would swing it into a distant orbit and make it a pinhead-sized moon. Among the rocks he hid the small, compact apparatus that could call it back when wanted.
The alien caused nearby humans to see it as a normal human but had limited range. Its actual appearance was humanoid enough to fool people who saw it from beyond its sphere of influence.
Holding a small case stuffed with notes and sketches, he studied the landscape, saw everything as it had been originally. To anyone within the sphere of his mental influence he was no more than a portly and somewhat pompous business man idly surveying the hills. To anyone beyond that range he was made vague by distance and sufficiently humanlike to the naked eye to pass muster.
But to anyone watching through telescopes and binoculars from most of a mile away he could be seen for what he really was—just a thing. A thing not of this world. They could have made a snatch at him then and there. However, in view of the preparations they'd made for him there was, they thought, no need to bother. Softly, softly, catchee monkey.
It didn't bother altering dogs' perceptions, it just made them scared so they ran away from it.
He ambled along the main street and was ignored by unsuspecting natives while practically rubbing shoulders with them. The experience gave him no great kick; he'd done it so often elsewhere that that he now took it for granted and was almost bored by it. At one point a dog saw him, let go a howl of dismay and bolted with its tail between its legs. Nobody took any notice. Neither did he.
This was noted by another character.
"They didn't see him using a car. He was on foot at the time and carrying a bag. They noticed and remembered him only because a mutt yelped and went hell-for-leather down the street. They wondered whether he'd kicked it and why."
It got into town by making a passing car-driver stop and give it a ride.
The first car that came along was driven by a salesman who never, never, never picked up a hitcher. He'd heard of cases where free riders had bopped the driver and robbed him, and he wasn't going to be rolled if he could help it. So far as he was concerned, thumbers by the wayside could go on thumbing until next Thursday week.
He stopped and gave Vanash a lift and lacked the vaguest notion of why he'd done it. All he knew was that in a moment of mental aberration he'd broken the habit of a lifetime and picked up a thin-faced, sad and silent customer who resembled a middle-aged mortician.
The driver later told the authorities about the incident, adding that he never picks up hitch-hikers.
"It was the craziest thing, captain. For a start, I never give rides to strangers. But I stopped and picked up the fellow and I still can't make out why I did it."
It bought supplies by making people think it was handing them money, but knew they'd eventually notice the missing amounts.
He sat on the edge of the bed and thought things over. It would have been an absurdly simple trick to have paid her in full without handing her a cent. He could have sent her away convinced that she had been paid. But she'd still be short twelve dollars and get riled about the mysterious loss. If he stayed on, he'd have to fool her again and again until at last the very fact that his payments coincided exactly with her losses would be too much even for an idiot.
When it wanted to be more discrete, it made a respectable-looking woman steal a wallet and dump it nearby. The woman then went home "and held her head."
Standing unnoticed at one side, he concentrated attention on a plump, motherly shopper of obvious respectability. She responded by picking the purse of a preoccupied woman next to her. Sneaking the loot out of the market, she dropped it unopened on a vacant lot, went home, thought things over and held her head.
The alien ended up being shot from outside its telepathic range, by a sniper using field glasses.
His immediate fear was well-founded. At twelve hundred yards there happened to be a beefy gentleman named Hank who found that a brazen escape during an outbreak of civil war was too much to be endured. Hank had a quick temper, also a heavy machine-gun. Seeing differently from those nearer the prey, and being given no orders to the contrary, Hank uttered an unseemly word, swung the gun, scowled through its sights, rammed his thumbs on its button. The gun went br-r-r-r while its ammo-belt jumped and rattled.
Despite the range his aim was perfect. Harasha Vanash was flung sidewise in full flight, went down and didn't get up. His supine body jerked around under the impact of more bullets. He was very decidedly dead.
The government found its landing craft and set about making copies, improving its weapons so they'd be ready for any possible invasion.
The hordes of Andromeda were very, very old. [...] Like many very old people, they had contempt for the young and eager. But their contempt would have switched to horror if they could have seen the methodical way in which a bunch of specialist legworkers started pulling their metal sphere apart.
Or the way in which Earth commenced planning a vast armada of similar ships.
A good deal bigger.
With several improvements.
