The whole passage of your quote can be translated as:
For that reason, it was almost only me who was able to speak to the people of the clinic. Though I admit I look a bit pert, I’m only a messenger girl of my master, and naturally have to deal with the doctors and staff of the clinic. It requires a bit of understanding of (medical) gibberish.
漢語 literally means Chinese language, but in a colloquial usage it means "difficult, technical words," because 漢文 was read and understood only by intellectual people in Meiji and Taisho era, not to mention earlier than those time.
In 永井荷風’s famous "墨東奇談” published in 1947, much later than this work, the heroine, Otsuru, a prostitute, whom the hero, Taneda, a school teacher describes as a beautiful crane among a bevy of chickens - 鶏群の一鶴 - tells him that her father was “an intellectual man who used to read “四角な文字” (square letters),” which means 漢字. You’ll understand that 漢字 looks square and Kana looks round in its shape.
There is also a Japanese word, チンプンカンプン meaning totally incomprehensible or nonsensical, which came from 珍文漢文 (ridiculous writing in Chinese language).
In net, 漢語 here means “gibberish” or Greek to laymen.