The story you described is "Dear Devil", a novelette by Eric Frank Russell, also the answer to this old question. The story was first published in Other Worlds Science Stories, May 1950, which is available at the Internet Archive. From a lineup of covers you picked Space, Space, Space as the anthology you were looking for. For the benefit of future users who may be looking for the same story, I will indicate the points where the story matches or differs from your description.
The best of the bunch involved a Martian who visited earth, and realized we'd blown ourselves up in a nuclear war and had fallen into savagery.
At first, the Martians can't understand what happened to Earth:
Poet Fander touched him, tentacles tip to tip as he sent his thoughts racing through their nerve-ends. "This confirms all that we saw as we approached. A stricken planet far gone in its death throes. What do you suppose caused it?
"I have not the remotest idea. I would like to know. If it has been smitten by natural forces, what might they do to Mars?"
The Martian is a vegetarian
"Good for dinner," remarked Speedy, touching him and speaking through the touch.
Fender felt sickened. Meat-eaters! It was not until a queer feeling of shame and apology came back at him that he knew the other had felt his revulsion.
who subsists on biscuits made by a machine he brought with him,
One day the eldest boy, who was named Redhead, came to Fander and grasped a tentacle-tip. "Devil, may I operate your food-machine?"
"You mean you would like me to show you how to work it?"
"No, Devil, I know how to work it." The boy gazed self-assuredly into the other's great bee-eyes.
"Then how is it operated?"
"You fill its container with the tenderest blades of grass, being careful not to include roots. You are equally careful not to turn a switch before the container is full and its door completely closed. You then turn the red switch for a count of two hundred eighty, reverse the container, turn the green switch for a count of forty-seven. You then close both switches, empty the container's warm pulp into the end molds and apply the press until the biscuits are firm and dry."
and communicates by using one of his tentacles to feed messages direction to/from the brain of the person he's talking to.
Captain Skhiva eyed his crew, spoke to them with his sign-talking tentacle. The alternative was contact-telepathy, which required physical touch.
Inadvertently left behind by his expedition,
No, he is left behind at his own request:
"Yes, Captain. I wish to be left here."
"What?—in this place of appalling repulsiveness?"
"It has an all-pervading air of ugliness and despair," admitted Poet Fander. "All destruction is ugly. But by accident I have found a little beauty. It heartens me. I would like to seek its source."
he initially befriended some children,
He beguiled them with his harp, teaching them to play, and now and again giving them ten-minute rides on the load-sled as a special treat. He made dolls for the girls, and queer, cone-shaped little houses for the dolls, and fan-backed chairs of woven grass for the houses. None of these toys were truly Martian in design, and none were Terrestrial. They represented a pathetic compromise within his imagination; the Martian notion of what Terrestrial models might have looked like had there been any in existence.
and eventually managed to rebuild civilization. Presently he dies, but when another expedition comes a generation later, they find the civilization he rebuilt.
Yes, except that he's not dead; he's in hibernation, and the Martians of the second expedition revive him.