This question is more complicated than it seems. Women's hair removal can be traced to ancient Rome, ancient Egypt, and ancient India. Through time and location, through differences in culture and fashion women shaving various body parts were popular/common. By mentioning your grandmother I'm guessing you are more interested in the last century or so.

For the last 150 years or so the answer is very simple: fashion, advertisements, propaganda, War, and health. The how is pretty much the same we use today, razors, wax, and depilatories.

With the change of hemlines and differing sleeve lengths more skin, and thereby hair, is on display. In 1915 Gillette debuted the 'Milady Décolleté'. Quotes within advertisements for this razor are: "[razor]... is welcomed by women everywhere-now that a feature of good dressing and good grooming is to keep the underarm white and smooth", "Fashion says- Evening gowns must be sleeveless... The Women of Fashion says- The underarm must be as smooth as the face.", and "Serves the modern woman who would appear at her best". Other brands with advertisements during this time convey the same message, 'if you don't shave your underarms you aren't clean, hygenic, pretty, fashionable'.

For 1923 I found an advertisement for 'ZIP' which a depilatory, I believe it's a lotion but the letters are small so I'm not positive. The ad has a painting of a young woman wearing a sleeveless bathing outfit reaching mid thigh, In large lettering it states "you need not be embarrassed!' and continues on with 'When you are going to the beach this summer, are you...afraid to raise your arm? ...shrink from the scrutinizing glance of your friends? Are you going to permit unsightly hair on your face, arms, underarms and limbs to spoil the freedom which awaits you at the beach?"

During the 20s and 30s shaving the legs was more for summer or as a personal preference due to stockings. At this point shaving the underarms was the norm and more companies and products (various adverts can be found) were being marketed towards women.

Then the 40's came and with it WWII. In 1939 DuPont introduced nylon stockings advertised as being lightweight, easy to wash, elastic, long lasting, and tough. However with the war came campaigns to donate nylon and silk stockings, nylon and silk were needed for soldiers parachutes and uniforms. Due to the lack of actual stockings, 'liquid' and 'paint' stockings(and their ads) came to be. 'Painted on' stockings however required shaved legs in order to have a smooth even finish that looks like silk. Liquid stockings weren't new though Elizabeth Arden's 'Velva Beauty Film' has multiple ads from the 30's. During wartime many tips for fake stockings included the use of eyeliner to make a seam for the stockings, I found various images pertaining to this but couldn't confirm the magazines they were from.

After the war stockings became available once more. In 1946 clothing designer Louis Reard introduced the bikini. In the 60's the miniskirt came to be.

I hope this answered your question, I'm on mobile so apologies for any mistakes and lack of links. All the advertisements referred can be found through google if you wish to see them.

There is certainly more that can be said, but I answered this question several years ago:

When did it become the social norm/standard for women to shave their legs completely?

I would note that I strongly question the standard popular narrative that leg-shaving correlates to the lack of stockings in WWII. As quoted in the previous answer,

By 1938, one expert could declare without sarcasm that any hair not on a woman's scalp was rightly considered "excessive".