Susskind corrects himself at 1:00:08 where he says that the "mass density" coefficient in front of the $\dot{X}^2$ term is a non-relativistic analogue; it is not actually a mass-density of the relativistic string. Instead it is proportional to the string tension $T$.

This is related to the fact that the non-relativistic string actually displays a Lorentz symmetry. However, the characteristic speed $c$ is then not the speed of light but the speed of sound, cf. e.g. this Phys.SE post.

To see how masses enter string theory, see e.g. this Phys.SE post.

To see how to go take the continuum limit in the discrete non-relativistic string, see e.g. this Phys.SE post.

Susskind corrects himself at 1:00:08 where he says that the "mass density" coefficient in front of the $\dot{X}^2$ term is a non-relativistic analogue; it is not actually a mass-density of the relativistic string. Instead it is proportional to the string tension $T_0$.

This is related to the fact that the non-relativistic string actually displays a Lorentz symmetry. However, the characteristic speed $c$ is then not the speed of light but the speed of sound, cf. e.g. this Phys.SE post.

To see how masses enter string theory, see e.g. this Phys.SE post.

To see how to go take the continuum limit in the discrete non-relativistic string (which is the main theme of Susskind's lecture), see e.g. this Phys.SE post.