How does slicer perform image compression similar to the Simplification: Quadric Edge Collapse Decision function of Meshlab to reduce the size of exported stl files?
Yes, we have several methods in Slicer that can provide equivalent or better quality mesh simplification and size reduction.
If you have only a few meshes to process then I would recommend to use Surface toolbox module’s Uniform remesh option (it creates very nice mesh with uniform triangle sizes) or Decimate option (it performs quadric decimation - probaby exactly the same as Meshlab - if you enable boundary deletion; or the slightly different DecimatePro algorithm if you disable boundary deletion).
If you need to process many segments (anatomical atlases, TotalSegmentator segmentation results, etc.) then I would recommend to use SlicerOpenAnatomy extension, which can apply quadric decimation to many segments or models at once.