Of American-English origin, bread-and-butter letter denotes a letter of thanks for hospitality, sent by a departed guest.

I have found an early instance in the column Just Among Ourselves, by Mrs Lyman Abbott, in The Ladies’ Home Journal (Philadelphia, Pennsylvania) of February 1894; the author explained why bread-and-butter letters were so named, why they mattered, and gave advice on how to write them:

I am glad to emphasize here the duty of acknowledging hospitality. Many people are very careless about it, especially young people, and they have less excuse than older people who may be neglectful under the press of business; care sometimes weighs so heavily that little things are forgotten. More than once I have been seriously troubled because a young friend has failed to acknowledge to me her safe arrival at her home after visiting me. The “bread and butter letter,” as it is sometimes called, because it is supposed to be an expression of thanks for what bread and butter stands for, should be written within twenty-four hours after arrival at one’s destination, to the hostess whose hospitality one has been enjoying. It is not quite enough for a young man who has been visiting his college mate to write to him alone; courtesy calls upon him to send at least a brief note to his friend’s mother, or the lady taking her place. You think it is not an easy thing to do, and it is not altogether if you try to make your note unique and different from others, but the simplest way is the best way, and if you have had a pleasant time say so. An agreeable incident of your journey, or a few lines about the circumstances into which you have gone, will make a letter which your hostess will enjoy, and she will set you down in her selected list of well-bred young people. Besides that, you will have the consciousness that you have been thoughtful of another’s feelings, and have not been guilty of the appearance of ingratitude.

An originally British-English synonym of bread-and-butter letter, Collins alludes to the letter of thanks for hospitality written by Mr Williams Collins, a character in Pride and Prejudice (1813), by the English novelist Jane Austen (1775-1817).

Mr Collins, vicar of Hunsford, is the cousin of Mr Bennet of the Longbourn estate; during his weeklong stay at Longbourn, Mr Collins has successively proposed marriage to one of the daughters of Mr Bennet, Elizabeth, who has refused, and to Elizabeth’s friend, Charlotte Lucas, who has accepted; his letter is thus mentioned in the novel:

The promised letter of thanks from Mr. Collins arrived on Tuesday, addressed to their father, and written with all the solemnity of gratitude which a twelvemonth’s abode in the family might have prompted. After discharging his conscience on that head, he proceeded to inform them, with many rapturous expressions, of his happiness in having obtained the affection of their amiable neighbour, Miss Lucas, and then explained that it was merely with the view of enjoying her society that he had been so ready to close with their kind wish of seeing him again at Longbourn, whither he hoped to be able to return on Monday fortnight; for Lady Catherine, he added, so heartily approved his marriage, that she wished it to take place as soon as possible, which he trusted would be an unanswerable argument with his amiable Charlotte to name an early day for making him the happiest of men.

I have found an interesting article about Collins in The Times (London) of Monday 23^rd February 1914; taking the opposite view to Mrs Lyman Abbott, the author explained why letters of thanks for hospitality were execrated and proposed to get rid of them altogether:

The Abhorred “Collins.”

A great many people will be sitting down to-day, after week-end visits, to write that letter of gratitude for hospitality which has come to be known as a “Collins.” There are some who have a gift for writing Collinses, just as there are politicians who can enthral an audience for an hour by telling them nothing, and journalists who can fill a column without having an idea in their heads. But to most the writing of a Collins is the sorest of all polite duties. To them the art is neither the gift of fortune, like being a well-favoured man, nor, like reading and writing, does it come by Nature. It is not so much the break-up of a pleasant party, nor even the bother of getting up early to catch the London train, that rolls Monday breakfast in gloom or lights it by flashes of hectic gaiety. It is the thought of the Collins that must be written ere the day is out. Shy young men have been known to decline week-end invitations rather than face the writing of a Collins. It is so difficult to find other ways of saying: “I enjoyed myself” or “I was bored to death, but do not want you to guess it.” The first Collins addressed to a particular hostess ought, like the first round at golf or the first attempt at archery, to be attended by “beginner’s luck”; it ought to seem quite easy. But it seldom does so. The writer is afraid of saying too much or too little; he is afraid of being too familiar or too formal; afraid of committing himself to friendship or of implying that he would prefer to stop short at acquaintance. And Collinses addressed to a hostess with whom he is on terms of friendship become more difficult with each visit. He wants to write to his friend; but not then, and not when a letter is expected of him. He has said this to her before, and that to some one else. This phrase is pompous, and that unworthy. Every heroic effort at the exceptional narrows the field for the future. To write a Collins in rhyme is to be condemned to prose for the rest of the bold author’s tale of years; and the more brilliant is this week’s Collins, the less hope there is for next week’s. The nuisance, of course, takes its name from the Vicar of Hunsford, and we are left to contemplate the significance of the fact that the Reverend William Collins’s letter on leaving Longbourn is not extant. It was “written with all the solemnity of gratitude which a twelvemonth’s abode in the family might have prompted.” That is all we know. Even Jane Austen shirked the writing of a Collins.
The writer is not the only sufferer; the hostess must share the trouble. We need not concern ourselves with the hostess whose parties are failures. She has not the gift of entertaining, and she will lack the sensibility to be affected by the Collinses. She will believe all that they say and never notice how they say it. But even hostesses to whom years of success have brought confidence have been known to feel uneasy on Tuesday mornings; while young and sensitive ladies have been plunged into undeserved misery. The guests seemed to have been so happy; there was so much talk and laughter; and the handshakes and smiles on parting did not speak of blessed release. Yet these letters—this so stiff and cold, that so much too effusive—set the young hostess wondering whether indeed her party was not a dismal failure. Only people, surely, who had been bored to death could express themselves so glibly or so lamely. It is but poetic justice, perhaps, that good manners should exact from the hostess some measure of the suffering which it forces upon her guests; but poetic justice is even clumsier than legal justice.
Life will be happier if during the weeks of the coming season a hint is taken from the history of calling after dinner. Within the memory of living man a visit in person was demanded; a hostess needed evidence that none of her guests was poisoned. Then the rule was relaxed, and it was enough if some member of the family left at the front door a piece of pasteboard printed with the name and address of the diner. This was not evidence, but society agreed to accept it as such. Next the family was relieved of the duty; a District Messenger or a taxi-cab driver was entrusted with the pasteboard; and, finally, even the pasteboard was excused. The first step from the Collins might be a printed form, with a blank left for the hostess’s name. There was once an undergraduate who derived much peace of mind from a store of forms printed as follows:—“Mr. A. B. presents his compliments to Mr. (or Professor, or Doctor) Blank, and regrets that he is unable to attend his lecture this morning.” The forms served their turn, and were succeeded by the shirking of lectures without either compliments or regrets; and the printed Collins would be, of course, but the prelude to a blessed silence.