The idea

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).

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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.

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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.

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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.