In Chapter 24 of Book the Second, Dickens writes with witty understatement,
Monseigneur, as a class, had dissociated himself from the phenomenon of his not being appreciated: of his being so little wanted in France, as to incur considerable danger of receiving his dismissal from it, and this life together.
Fearing for their lives, the aristocrats of France have fled across the English Channel. Since Tellson's Bank has both a Paris branch and the bank in London, some of these aristocrats, who have come to England, have transferred their funds to the London bank. But, even the ones who are "without a guinea haunted the spot where his guineas used to be."
As was natural, the head-quarters and great gathering-place of Monseigneur, in London, was Tellson's Bank....Moreover it was the spot to which such French intelligence as was most to be relied upon, came quickest....Tellson's was at thtat time, as to French intelligence, a kind of High Exchange....so numerous that Tellson's sometimes wrote the latest news out in a line or so and posted it in the Bank windows, for all who ran through Temple Bar to read.
Because Tellson's Bank has served these aristocrats in Paris, it extends courtesy to them in London. In addition, those who have lost their money know that the wealthy aristocrats have funds in Tellson's, so they come in hopes of charity. In addition, it is the place of sanctuary for the aristocrats where they commiserate and learn information.
Of course, this setting is one which delights the pretentious Stryver who shoulders his way among the Monseigneurs who discuss the state of affairs. It is at Tellson's Bank that the letter for Charles Darnay arrives in hopes that among the aristocrats who gather there, Monsieur d'Evremonde will be there to receive the desperate missive of Gabelle.