The Australian and New-Zealand phrase to frighten (also to scare) seven bells out of somebody means: to terrify somebody.

This phrase is probably modelled on the earlier to knock seven bells out of somebody, which means: to give a severe beating to somebody.

The phrase to knock seven bells out of somebody occurred, for example, in the following two texts:

1-: From the account of a public inquiry into the treatment of the passengers on board the Indian, an emigrant ship, published in the Supplement to the Adelaide Observer (Adelaide, South Australia, Australia) of Saturday 15^th September 1849 [page 2, column 3]:

Mr Bainbridge described a similar outrage committed on him by the Captain’s clerk since they arrived in port. He was called up out of his bed after he had returned from town wearied looking for employ and a house to remove to. On coming on deck, he was knocked down by Mr Ross, and the captain wanted him or any of them malcontents to stand before him, and he’d “knock seven bells out of them.”

2-: From Local and General, published in The Akaroa Mail, and Banks Peninsula Advertiser (Akaroa, Canterbury, New Zealand) of Tuesday 3^rd May 1881 [page 2, column 6]:

Yesterday evening Mrs Burns gave her interesting lecture on “Tom o’ Jack’s Lad” in the Congregational Meeting-house here, and for the benefit of the funds of the congregation. The lecturer, without any pretensions to literary finish, was terse to the point, showing how it was possible to make the best of both worlds. Tom o’ Jack’s lad, the Lancashire pitman, after knocking the colonial seven bells out of all his mates, embraces Methodism, opens a store, succeeds in business, and dies Mayor of his native town, worth £9000.

The earliest occurrences that I have found of the phrase to frighten (also to scare) seven bells out of somebody are as follows, in chronological order:

1-: From the Taranaki Herald (New Plymouth, Taranaki, New Zealand) of Thursday 1^st May 1890 [page 3, column 1]:

STRATFORD AND NGAIRE.
[FROM OUR OWN CORRESPONDENT.]

April 30^th.—[…] What at first appeared to be some horrible mystery occurred last week. As a settler was wending his way home along the Cheal Road, about 1 a.m., on a frosty night, he was horrified at hearing some muffled moans and shrieks, from someone apparently in mortal agony. As he stood motionless, trying to trace the direction of the sounds, he suddenly saw a ghostly female figure, carrying a light in one hand, emerge from the surrounding gloom and pause over what appeared to be a deep and newly dug grave. Then another ghostly form appeared, also clothed in white, and advancing to the same spot lowered what seemed to be a ladder into the grave, from whence continued to come the same mysterious and blood curdling sounds. Then the figure of the second spectre slowly descended the ladder, and shortly emerged, bearing in its ghastly embrace, a struggling, moaning, object that looked like a resuscitated corpse. The horrified spectator was so overcome by terror that he fled from the spot, and did not pause a second until he had reached the security of his own home. As he was about to communicate with the police on the following morning he accidentally received a solution of the mystery. It appeared that a neighbour of his Mr G. C., had a valuable pointer pup that, somehow, during the night, had managed to tumble into the well. Aroused by its cries, he sprang up in his nightshift, and finding out the accident, and that no time was to lost [sic] if he would save the dog from drowning, aroused an old lady who occupied the same house, and got her to come at once in her nightdress, and hold a light for him whilst he descended the well and rescued the pup. The white colour of the pup, and the night habiliments of the performers, and other surrounding circumstances, had tended to create such a complete ghostly illusion as to frighten seven bells out of one of our sturdiest and most courageous pioneers of the backwoods. As the poet says, “What great events from seeming trifles spring.”

2-: From Spontaneous Combustion, by ‘The Dipsomaniac’, published in The Bulletin (Sydney, New South Wales, Australia) of Saturday 4^th April 1891 [page 10, column 3]:

I speedily became a Shining Light, used to speak at open-air and hall meetings on the curse of drink and the blessings of Gospel Temperance […].
[…] I was particularly happy in impromptu efforts at frightening “seven-bells” out of the most hardened boozers, by vivid descriptions of the torments of a drunkard’s hell, both in the earthly fantods and the after-death eternal drought. It was a common thing for me to harrow a trembling sly-drunkard into jumping horrors, and bring him to the penitents’ form in delirious fear.

3-: From the column Jottings, by ‘Batsman’, published in The Sportsman (Melbourne, Victoria, Australia) of Tuesday 16^th October 1894 [page 6, column 2]:

South Melbourne, like North Melbourne, have had troubles of late. Dr. E. Barrett and Graham are with them not, and they have had to cast around for new blood. And they have found it, and I might add some old. Dr. Mailer, Ellis, Tatchell, and Palmer are now in the team, which looks quite a formidable bunch of players. In the game against Fitzroy commenced last Saturday at South Melbourne, at any rate for the first few minutes, they looked a precious weak lot, for Mitchell sent Trott, Bowman, and Dr. Mailer to the pavilion in quick succession, and had frightened “seven bells” out of the team generally. Three wickets for one run! Think of that; and two of the victims were batsmen who, the previous week, had compiled the century in great style.

4-: From Tales of the Tender Passion. Jim’s Last and Serious Case, by the explorer, journalist, historian and fiction writer Ernest Favenc (1845-1908), published in the Evening News Supplement (Sydney, New South Wales, Australia) of Saturday 1^st August 1896 [page 2, column 1]:
—Context: A coachman is telling the story of one of his trips; among his passengers were a friendly school mistress and a finicky parson; the parson was inside the coach, and the school mistress on the box seat:

The school mistress felt a bit sorry for the parson, and she said she would take a spell inside for a while, and let the parson take her place. So up he came, and you bet I raced down the ridges, and frightened seven bells out of him.

5-: From A Buccaneer’s Diary. Part II. The Recital Continued, and the End of the Treasure, by the explorer, journalist, historian and fiction writer Ernest Favenc (1845-1908), published in the Evening News Supplement (Sydney, New South Wales, Australia) of Saturday 16^th April 1898 [page 3, column 4]:

Ram Mung returned for what he had buried, and took the crew of his proa ashore to dig it up, and carry it on board. When they had got it, the crew knocked old Ram Mung on the head, and tumbled him into the hole, all standing. Then they departed, and Mordrassa frightened seven bells out of them, and they all came to grief. Now, Mordrassa is amusing himself at the expense of anyone who comes looking for the treasure. He’s just the sort of ghost to play a mean trick of that sort.

6-: From the column In the Bar Parlor, published in The Clipper (Hobart, Tasmania, Australia) of Saturday 14^th May 1898 [page 6, column 1]:

“So the Flying Dutchman has popped up again,” remarked Jim the Luney.
“Nothing new about that, surely?” queried the Sunday Man. “Hasn’t he been with us for years?”
“Rats!” exclaimed the Sporting Man. “What are you talking about?”
“Why, the Flying Dutchman of course. Don’t you mean Doctor ——”
“No,” yelled Jim the Luney. “Didn’t you ever hear of the phantom ship that used to act as a sort of Phosphorous Jack to sailors in ye olden time—a mysterious “prepare-to-meet-your-doom” craft that sailed noiselessly around in troubled seas and frightened seven bells out of peaceful mariners?”

7-: From one of the unconnected paragraphs making up the column “Pars” about People, published in The New Zealand Observer and Free Lance: An Illustrated Journal of Interesting and Amusing Literature (Auckland, Auckland, New Zealand) of Saturday 10^th September 1898 [page 6, column 4]:

The Maori lunatic scared seven bells out of the Remuera residents last week. It was worth a good deal to see several of the aristocrats of the aristocratic suburb careering after the nude warrior, who sprinted up and down the sands like an antelope.