First of all, keep in mind that Armande is a character who will do what she wants to do, despite what others think (a fact which infuriates and eventually alienates her daughter).
It seems, therefore, that Armande actually sees similar qualities in Vianne when she opens up her Chocolaterie in a small town right in the middle of Lent. You asked first in your question "how." Well, Armande comes to the shop, and she and Vianne click immediately. Through what become fairly regular visits to the shop, Vianne and Armande seal a wonderful and open friendship. First, Vianne discovers who Armande's daughter is (a very religious and pious woman who is ashamed of her mother) and then that Armande does not have a relationship with her grandson. Vianne arranges for the two to meet once a week and read poetry. It is almost as if Vianne is the daughter Armande never had and always wanted.
She discovers that Armande has a history with the priest in town who is laying guilt on all his parishoners for their Lenten indulgences. Armande does not attend weekly mass and makes no excuses for it.
Eventually, and probably too late, Vianne discovers that Armande suffers from (presumably as I don't believe it is ever mentioned by name) diabetes, and the indulgences she affords herself in the Chocolaterie are probably killing her. Not that knowing would have mattered. Armande is a woman who does what she pleases despite and sometimes to spite anyone who gets in her way.
In the end, the entire family comes together (with others) to celebrate Armande's birthday. Armande herself has planned the whole thing, of course, but it could not have been carried out if it were not for Vianne. In this way, Vianne is responsible for bringing Armande together with her grandson and her daughter - but in doing this - also reconnects Armande's grandson with his mother.