It's not compaction, but moisture conditioning the expanisve soil which will reduce the expansion potential. Expansive soils swell when water is introduce. The theory is that if the expansive material is excavated, broken up to clumps smaller than 2" or 3", moisture conditioned to optimum or even a little above, and then compacted. This should result in a less expansive soil deposit within the zone of moisture variation. This has to be done by a contractor experienced in working with expansive soils and compacting material at or above optimum. The soil shouldn't be left to sit uncovered for long though. The expansive soil could dry out, shrink, and regain some or all of it's expansion potential. If the site is left to dry out too long, the contractor might have to redo some of the moisture conditioning.
After the fill is in place, it is prudent to drill more borings to test the fill after this is done to make sure the contractor did it correctly (or the field technician didn't use the wrong proctor/opt. moisture content).
Conversely, compacting the soil will make it MORE expansive. It's a balancing act as you need to compact the soil to reduce settlement but also don't want to over compact the soil which can make it more expansive. The trick is not to over compact the upper portion of expansive fill. I've seen specs of 95% of the standard proctor instead of the modified proctor for any moisture conditioning and compaction of expansive soil within 10' or 15' of a foundation. Any fill deeper than 10' or 15' should be compacted to a higher degree, i.e. 95% of the modified proctor to limit settlement concerns.
I should note that this method is relatively new and not used by a lot of engineers or on a lot of projects. I've only seen this used for large residential projects with basement slabs that the owners accept there could be some movement and distress in the basement slab. MEP lines at transition points are typically flexible.