It’s pretty clear when you see the those classic pink eye symptoms that you’ll be needing some drops to treat this very contagious infection, of which there are some three million cases a year, according to the Mayo Clinic.

the和those都是中位限定词，中位限定词互相排斥，一个名词词组最多只能用一个中位限定词。因此应删去定冠词the。句首的it显然是形式主语，后接that引导的真正主语从句。从逻辑上看，when从句应该是主语从句中的时间状语从句，因此that应该位于when之前。虽然在使用形式主语it的情况下，that引导主语从句有时可以省略that，但如果主语从句比较长，则不宜省略。这个句子修改后如下：

It’s pretty clear that when you see those classic pink eye symptoms，you’ll be needing some drops to treat this very contagious infection, of which there are some three million cases a year, according to the Mayo Clinic.

一个句子出现两处明显错误，实属不应该。可能是网络上有人缩写文章导致的。