I suspect that the reason to use it is that the cans and the associated empty space is cheaper than actually having solid concrete (and may be partly necessary to reduce the weight of the concrete and thus associated stresses with it).
You're probably (mostly) correct. For beams, reducing the weight of the beam itself will reduce stresses in the beam and the members supporting it, as well as reducing the material cost of construction.
However, for columns, reducing weight is not necessarily a good thing. Columns are influenced by what's called P-M behavior, which is the combination of axial load and bending moments on a column. In general, if one increases the axial load on a concrete column, one also increases the bending capacity of that column (until one of these parameters hits a critical value). The image below shows a simplified P-M curve, and you can see that increasing axial load increases moment capacity up to about midway up the curve.
If you remove part of a column's cross-section, you remove it's axial capacity, which is not a good thing.
Also, you'll notice that there are little layers of concrete between the cans. Ideally, one would want these larger so that the "webs" of concrete can transfer force between the layers. Having essentially two separate layers of concrete ~3" thick isn't strong at all.
However, the 'Styrofoam and foamwork boards' being the acceptable standard now makes me wonder how different that actually is from empty cans.
In practice, there's probably not a huge difference since both introduce a void in the concrete. Styrofoam might be a bit better since it won't deform as much as a tin can and thus hold up to the hydrostatic pressures of concrete (during a pour) better. Also, it's probably easier to get styrofoam that is manufactured to a controlled process than it is to get consistent tin cans.
What you end up with is something like a waffle slab, just that the holes in the waffle are interior to the slab.
As an aside, the engineer in me really recoils at the thought of doing this to a concrete slab. If I saw a contractor doing that to one of my designs, we would have problems.