The elephant's arms and legs were removed and used for arm warmers and leg warmers, and his little gray tail attached to the back of a grey track suit. (It sounds gory, but try not to imagine real elephants on the savanna. Concentrate on the cute pictures and we'll get through the alterations just fine.)
Possibly the best part (for me, because I don't have kids) is that this costume isn't necessarily just for toddlers. Riet says it even fits an adult (presumably with a different size track suit), so if you can deal with a few weird looks you can have this much fun, too.
Kapplar the cuddle elephant seems to be discontinued, but a few still roam the Internet, and this technique seems like it could work for any stuffed animal of suitably elephantine size. I'm never going to look at a large-size stuffed animal the same way again.
(Images via Leven met Liv)