What is the explanation from the structural behavior and material micro structure point of view for, why the compressive fatigue strength of a material is more than that of a pure tension or fully reversed cyclic load?
I think your question is why compression-compression fatigue life is more than tension-tension or tension-compression (fully reversed) fatigue life. The possible reason is as follows.
You may be aware that fatigue is a surface phenomena in which the back and forth movement of dislocations (upon cyclic loading) along slip planes/slip directions will lead to formation of persistent slip bands-PSBs ( a kind of slip traces that are permanent). These PSBs are essentially an extra plane of atoms that upon reaching the specimen surface produces small permanent projections, called extrusions (there are theories that say complimentary intrusions....a sink-in of plane of atoms often forms). The extrusions upon continuous cyclic loading will intensify and eventually create a surface imperfection (notch effect) and a micro crack will finally initiate from the intrusion/extrusion....that's is the theory of how a fatigue crack will initiate...
If your cycle is tension-tension or tension-compression, you are forcing the dislocations to move outward, sufficient enough to create intrusions/extrusions after a reasonable number of cycles. On the other hand if your cycle is compression-compression, dislocation movement is largely inwards (barring a few dislocations that may go in the other directions). The possibility of formation of intrusion/extrusion....the primary phenomena that controls the fatigue life...is ceased or at least delayed substantially. That is the reason why you would find longer fatigue life (higher fatigue strength as you put it) in compression cycling. How would you define fatigue failure in such cases is a challenging issue...