In Colombia it means siding for overconfid confidence. Believe that after you get a goal you don't have to keep fighting (at least to hold on).
In this locution the laurels represent the glory, the honors achieved; so falling asleep in them is relying too much and not continuing to strive, which can lead to losing the position achieved. See Dafne .
rest on the laurels is incorrectly written and it should be written as "Rest on their laurels" being its meaning: Falls asleep on its laurels the person who let go of the opportunity to obtain a benefit, by being in a State of comfort