It preserves the property that a problem has a polynomial time approximation scheme (PTAS) and is used to define completeness for certain classes of optimization problems such as APX.
With ordinary polynomial-time many-one reductions, if we can describe a reduction from a problem A to a problem B, then any polynomial-time solution for B can be composed with that reduction to obtain a polynomial-time solution for the problem A.
[1] Formally, we define a PTAS reduction from A to B using three polynomial-time computable functions, f, g, and α, with the following properties: From the definition it is straightforward to show that: L-reductions imply PTAS reductions.
As a result, one may show the existence of a PTAS reduction via a L-reduction instead.
[1] PTAS reductions are used to define completeness in APX, the class of optimization problems with constant-factor approximation algorithms.