Basically, the whole point of calling itself as an Empire is to protect the independence and identity of Korea.

The late 19th century was a really crazy time for Joseon (조선,朝鮮).

In 1894, there was the Donghak Revolution(동학농민운동, 東學農民運動 ), which mass protests by farmers of the south. To halt this, the Joseon government requested Qing to send troops. Due to the Treaty of Shimonoseki, the Japanese also came in with troops. The two militaries clashed, and the First Sino-Japanese War broke out inside of Joseon. And well, the Japanese won this.

Meanwhile, Russia wasn't happy that Japan took the Liaodong Peninsula since they were planning to build a port over there. The Triple Intervention happened and Japan gave up that peninsula. Queen Min and the Min Family (the clan that pretty much had a heck ton of power during King Gojong's reign) became very pro-Russia and the Japanese weren't happy about this. And they literally assassinated Queen Min and warned King Gojong. Gojong pretty much got scared and ran over to the Russian embassy and stayed there for a year.

People and groups like the Independence Club(독립협회,獨立協會) continued to request Gojong to come back to the palace and in 1897, he finally did. As he came back, he announced that he's gonna modernize and westernize the nation, and make it resistant to foreign threats.

This is called the Gwangmu Reforms(광무개혁, 光武改革), and it's pretty much "him saying that he's the emperor now".

There was no Emperor status in Korea during the Joseon period. Basically Korea acted as if it were a servant to the Chinese (Ming, and later Qing). Therefore, the only true emperor that exists was the one in China. This kind of relationship is called Sa-dae(사대,事大).

Gojong announcing that he's now the emperor means that Korea is no longer a servant state of China. It is a fully independent state that can function on its own. Which was pretty important, since 1890's Korea was a place where Japan, Qing, Russia, and other western powers were having "great interest" on.

Since Korea is now under an emperor that has all the power, the country is pretty much (in theory) an empire. The name "Korean Empire" is "Daehan-jaegook/대한제국(大韓帝國)" in Korean. The 제(帝) part means the Emperor, thus meaning that the name kinda literally means "Country(國) of Korea(大韓,Big Korea) ruled by an Emperor(帝)". That's why when translated into English, it's "Korean Empire".

Additionally, if you take out the 제(帝) and replace it with a 민(民), it becomes the "Republic of Korea", since it is literally "Country(國) of Korea(大韓,Big Korea) ruled by its people(民)". This is kinda where the official name of South Korea came from.

I skipped a lot of parts but this would probably be enough to answer your question. Hope that this helps.