The explanation to this is very simple: The Chinese Han and the Korean Han share different Chinese characters (Hanzi/Hanja) that refer to different peoples. In fact, as you study classical Chinese, you'll find that there are many variation of Han that sound the same when it is romanized, but are in fact different characters with ever so slightly different intonations.

Korean Han is 韓. This is in contrast to the Chinese Han 漢, which we still use to refer to Chinese characters 漢字. It originates from the Samhan period where three confederation of Korean tribes resided in the southern half of the current borders of the Korean peninsula. The three tribes: Byeonhan 弁韓 Jinhan 辰韓 and Mahan 馬韓 share the same character used by Koreans for their Han.

大韓 Daehan is the combination of the character for "Great" and the character for the Korean Han. It is the official title of our country 大韓民國 which simply says this is the nation of the Great Han (Korean) People. 大 was used historically to indicate sovereignty, in a way that's similar to the Venetian Republic's use of serenissima. You'll find many chinese dynasties use the same like 大明 for the Ming dynasty for example.

In Korean context, you many also hear the word Han as a cultural feeling of pain and regret which uses the character 恨. This is pronounced hen in Chinese.

The confusion between Chinese and Korean Han has persisted to modern day in fact. The use of the Chinese Han 漢 is also used for the Han river 漢江 that runs through the middle of Seoul, South Korea's capital. Historically, Seoul was known as 漢城 which was derived from the river's name to mean that "Castle on the Han River". However, this was sometimes mistaken as the Chinese Han so the name was changed to the pure Korean name of 서울 Seoul (which had been in concurrent use since the 1800's but entered exclusive use after liberation), which just means capital city. The new Chinese character used is 首爾 Shou'er, which is a transliteration of the new name.