There were many reasons why people reacted as they did, some already have been mentioned.
The main reason was that she was the best thing that had happened to the Royal Family when she married Charles.
Before Diana the Royals were seen as very stuffy, unemotional and only interested in horses and horse shows. She brought a huge breath of fresh air to the whole family with her caring ways, her beauty, her youth and the fact that she had normal feelings towards her children and husband, like the rest of us. She cuddled them, she hugged them, she kissed them in public and she cared about them and loved them. They went to school near to Windsor, not hundreds of miles away in Scotland, as Charles had done.
She was privileged and she knew that and acknowledged it. She wanted to use her privilege and fame for good by supporting often unpopular causes, like Leprosy, Aids and Landmine charities. She also did many small kindnesses and was a genuinely caring person.
She was treated very badly by Charles and those around him. He knew he loved someone else when he married her and should never have done so. When it was clear she was suffering from feeling unloved and from all the attention from the media, then there were stories put out that she was "mad" and she was even called a "loose cannon", not long before her death when she was in Africa attending an event for a landmine charity.
She was probably the only Royal who the public could really connect with. She had wealth, privilege and fame and everyone loved her. The only one who didn't was Charles. Many women could understand her situation - her favourite song was "I will survive"!
Don't forget 1997 was before the internet, facebook, emojis and it was still a "stiff upper lip" time. People didn't hug in the streets like they do now. I think Diana made us more emotionally aware and mature and not afraid to express our emotions - that will be one of her legacies. She was one of a kind and I still feel hearbroken at her death, especially for her sons and her grandchildren who she never met. It's important to remember her as she was and to understand why she was so loved by ordinary people.