What did Mr. Nuttel see when he looked out of the open window? What do you think he felt and how did he react?

The author describes exactly what Framton Nuttel saw when he looked through the open French window.

In the deepening twilight three figures were walking across the lawn towards the window, they all carried guns under their arms, and one of them was additionally burdened with a white coat hung over his shoulders. A tired brown spaniel kept close at their heels. Noiselessly they neared the house, and then a hoarse young voice chanted out of the dusk: "I said, Bertie, why do you bound?"

Because of "the deepening twilight," the three figures would conceivably look somewhat "ghostlike," as if they were transparent and shimmering. Vera had prepared Framton to recognize them as the three supposedly dead hunters. They are all carrying guns. One of them has a white coat hung over his shoulders. Vera told Framson:

Poor dear aunt, she has often told me how they went out, her husband with his white waterproof coat over his arm, and Ronnie, her youngest brother, singing "Bertie, why do you bound?" as he always did to tease her, because she said it got on her nerves. 

Bertie's idiotic song will get on Framton's nerves, too. Vera also mentioned the "little brown spaniel" that was lost with them. All these distinguishing clues make Framton sure these must be the three dead men who have somehow managed to free themselves from the bog after three years and are returning for tea, just as Mrs. Sappleton expected.
The author has created the perfect victim for Vera's practical joke. Framton is suffering from what we would now call a severe neurosis. He is only present at the Sappleton household because he has come down to the country for a "nerve cure." It is because he is so nervous that he responds so vigorously. He is terrified. The fact that the three "ghosts" are all carrying guns makes them even more frightening. Ghosts would be bad enough; ghosts carrying guns are even worse. 
Vera has twice been described as "self-possessed." What triggers Framton's extreme reaction to the approaching hunters is the girl's feigned look of horror. She loses all her self-possession and appears to be seeing the same men her supposedly demented aunt has been expecting for three years. Everything works out perfectly for Vera's joke. Bertie breaks into song. Her aunt says everything she was expected to say.

"Here they are at last!" she cried. "Just in time for tea, and don't they look as if they were muddy up to the eyes!"
Framton shivered slightly and turned towards the niece with a look intended to convey sympathetic comprehension. The child was staring out through the open window with a dazed horror in her eyes. 

The reader isn't let in on the joke until Framton has gone running for dear life up the country road. Just like Framton Nuttel, the reader is beguiled into believing the three men approaching the open window in the deepening twilight are ghosts. It isn't until Mr. Sappleton enters through the window that the reader realizes it was all a set-up for poor Framton Nuttel.

"Here we are, my dear," said the bearer of the white mackintosh, coming in through the window, "fairly muddy, but most of it's dry. Who was that who bolted out as we came up?"
"A most extraordinary man, a Mr. Nuttel," said Mrs. Sappleton; "could only talk about his illnesses, and dashed off without a word of goodby or apology when you arrived. One would think he had seen a ghost."