I agree with AndreJ - I've used scapetoad for a few years and it works really well (but slowly, for good results)
However, with the cartogram plugin, check out the "add demo layer" menu option. It adds a layer showing Danish regions. Running on this gives a valid result
I noticed that the demo uses a local UTM projection in meters. I think this is key. You're using a projection in degrees.
I was able to get a worldwide example to work, using the same example as AndreJ, but by reprojecting the shapefile to EASE Grid (epsg:3410) to get coordinates worldwide in meters, and using MapColour1 as the value (which has a fairly small range, 1-10 or so)
That's a hacky way to do it and i don't recommend it as the distortions will mess with the areas of the cartogram, but it does suggest the cause...
I think that the ratio of the area to the value is key to getting this plugin to work.
I think you might need to experiment with reprojecting/scaling the polygons and/or scaling your values up or down to get a decent result.