I took a seat at the end of the hearthstone opposite that towards which my landlord advanced.

= I took a seat at the end of the hearthstone opposite what my landlord advanced towards.

what=that which。由于有的人不喜欢用介词结束一个句子，所以将what改为that which, 然后which作介词towards的宾语，towards which引导定语从句修饰that, 并在从句中作advanced的方向状语。

这种用法是相当正式的。