Wait - what? How does injecting insulin make you LOSE weight?

My understanding was that insulin sends glucose to the cells, making you fatter. Having a lot of excess insulin (hyperinsulinemia), or injecting too much insulin sends more glucose to the cells, making you fatter.

Injecting way too much insulin can result in death though, if you inject so much that your blood sugar goes way too low.

(Seriously, if excess insulin made you lose weight, I should be thin as a rail, since I've always produced excess insulin. I just always got hungry when my blood sugar dropped, so I ate and brought my blood sugar back up, producing more excess insulin, making me pudgier and pudgier as the years went on)