Why is the weight of the hanging part of the chain equal to the friction on the remaining part of the chain kept on the table?

Consider the horizontal forces acting on the part of the chain kept on the table. There are two: friction $F_f$ (acting to the left) and tension $F_T$ (acting to the right). For the forces to be balanced, it is necessary that $|F_f| = |F_T|$.

Now let's consider the vertical forces acting on the hanging part of the chain. Say this part of the chain has mass $m_2$. There are two forces: gravity $m_2g$ (acting downwards) and tension $F_T$ (acting upwards). For the forces to be balanced, it is necessary that $|m_2g| = |F_T|$.

Why do these tensions have equal magnitude? Essentially, the connection between the two parts of the chain "redirects" the tension force along the direction of the chain.

Thus $|F_f| = |m_2g|$.