
Alright so I'm a total history nerd, I majored in East Asian Studies and eat stuff like this up.
Read the full (long!) article here - I will try to explain it basically, but its really worth the read.
Here we go:

It all started with Japan.
As you might know, Japan colonized Korea eventually trying to erase Korean culture all together on the peninsula. In the very beginning though, that wasn't the case. Japan was willing to share Korean and Japanese culture, and the Japanese and Korean elite often spoke together and traded pop culture.
Japanese elite brought over Western-influenced music because of their connections with the Western world (which Korea didn't interact with other than missionaries at the time) The Korean elite wanted to be modern and successful like Japan, so they adopted this music (called Changga) as their new favorite genre. (You can hear a lot of European influences in early Korean music...)

This is where it gets interesting.
The Korean elite wanted to be independent of Japan. They wanted Korea to stand alone as a strong nation, not pushed around by China or Japan anymore. They stared the pro-independence movement.
Now, most revolutions like this would turn to traditional music, traditional culture to strengthen the country (for example, Korea would use ancient Korean songs to protest the Japanese occupation) but the Korean elite didn't! They wanted Korea to be strong, and to be strong they felt like they needed to modernize - ie be like the West. They used this Changga music (ironically brought to them by Japan) to protest Japanese rule!
To this day Korea is always trying to "keep up with" the Western world so that they can be considered modern and powerful. Most things that they do is to modernize and stay trendy because that is how the country has always defined strength.

But then, Japan decided to make money off of that...
Japan had seen its first real pop music hit in 1914. People were buying records, following tabloids, and really getting into pop culture. That was really the birth of the Japanese music industry.
In 1926, Korea had a similar experience. A song called the "Death Song" appeared, sung by Yun Shim-Deok. The singer committed suicide with her lover right after the song was released and the story made HUGE headlines. It was chilling, 'romantic,' and dramatic. Korea ate it up. Japanese record companies came in and started selling the record, selling over 50,000 copies! (In 1926 that number is totally insane!)
You can listen to the death song here:
The record industry after that was run mainly by the Japanese yakuza and their partnered Korean gangsters.
The industry was set up very similar to how it is now, with entertainment industries doing everything in-house (SM has specific song writers that only work for SM, and their artists are actors, singers, models, etc)
This is also why so many Kpop artists become actors...because even back in the 1920s musicians couldn't make enough money so the real way to make a living was through acting. That still stands today.

So, today's Kpop is based off of this first "Korean pop" hit (the death song) AND protest songs against Japan.
So when people say that Kpop is just a copy of Western music, they're sort of right but not in the way that they're thinking. Yes, its a reflection of Western culture, but the reason behind it is that Korea has been fighting to be recognized as a modern, powerful force to be reckoned with since the early 1900s.
They want people to know Korea is strong, modern, and a country to look up to, not look down upon.
So the next time someone tells you Kpop is a lame copy of American pop - give them this little history lesson.