The Mix 'Add' mode is combining your renders without taking account of which of the two images is 'in-front' of the other. Blender needs to know how to combine the two images correctly.
Looking at your screenshot the Render Layers thumbnail shows you aren't using transparency in your renders (it's a solid background instead of checkered). The simplest solution is to enable Transparency :
Then once you have re-rendered your scene you should have an Alpha channel in your render results (the background should be transparent). You can then simply use the Alpha Over node instead of 'Mix' to produce the 'overlay' effect you want :
An alternative (if the depth information is consistent between your render layers) is to use the Z-Combine node to combine two images based on depth (this allows you to have different elements from multiple render layers overlapping each other) :