Short answer? It was a fashion statement. It's almost like asking "Why did men in the 18th century wear knee britches or those big silly wigs?" It was just what caught on. Like many aristocratic fashions, the walking cane was born out of courts of Europe as a fashion accessory. There are famous portraits of monarchs like King Henry VIII, King George II & III having portraits done with a cane; not because of weakness but as a symbol of authority. King Louis XIV of France carried an elaborate, jewel encrusted cane and actually forbid his subjects from carrying a cane in his presence. It was a symbol of his power. In late 17th century and early part of the 18th century, the cane began replacing the sword as the accessory of choice for gentlemen about town in Europe and in the colonies. The more decorative a cane was, the more wealthy the gentleman was and it could still be use as a weapon in dire circumstances.

Just because it was a fashion statement didn't mean that there wasn't a practical use for it. For instance, doctors were well known for carrying a cane. Vinegar was believed to ward off illnesses so many would have a hollowed out head with a vinegar soaked sponge in it. The doctor would hold the head of the cane in front of his nose and inhale the vinegar as he visited patients, kind of like a protective mask. "Gadget canes" also became popular with doctors as they would use hollowed out canes to store their medical devices and drugs when making house calls. This allowed them to draw less attention to themselves and lessen the chance of them being robbed than if they were carrying a medical bag.

Edited a couple times for clarity. I'm tired and on my phone

Edit #2: I should be asleep right now, but instead I'd rather make lengthy posts here. I've lost control of my life...

Adding onto this: were cane swords ever used as a serious weapon?