The idea is quite simple.
Assume that one starts with a parameterized curve such that the speed is
a continuous function of time, and is never equal to zero.
Integrating the speed (indefinite integral) gives one the distance travelled as
a function of time. The usual symbol is s(t).
Since the speed is always positive, the function s(t) is strictly increasing, and
hence invertible. Solve for t in terms of s, and substitute back, similar to the
general reparameterization above.
The problems occur in two places -- the arc-length integral is "notoriously
impossible" for almost all nontrivial curves. Moreover, even if one can get
a formula for the integral, one very rarely can solve the resulting equation.
In practice, one usually gets by without formulas, using only numerical tables.
Nonetheless, for theoretical work "parameterizations by arc-length" are extremely
important and useful.