Well, since Chinese characters have been used for thousands of years, some aspects of the Chinese cultural heritage are embedded in the writings, which were comprised of Chinese characters. If people learn Classical Chinese, they could tap into a larger cultural treasure trove.
I mean, you coud read these ancient writings in the original language and feel what those people wanted to express. So, if you learn Chinese characters, you could learn both modern Chinese and Classical Chinese. I guess it’s like a Westerner who could read English and Latin.
Here is an example. This is a photo of a work, Duobao Pagoda Stele, or 多宝塔碑 in Chinese, by the renowned calligrapher Yan Zhenqing, or 颜真卿 in Chinese, who wrote in 752, which was in Tang Dynasty. Chinese People of nowadays and foreigners who learn Classcial Chinese could also read this and marvel at this masterpiece by an eminent maestro of the Chinese calligraphy.
As a matter of fact, his calligraphy is also appreciated by Japanese people, who made an exhibition of his works in 2019 at Tokyo National Museum. Here is a poster about this exhibition.