As far as I can see, all I would need to do would be show up on polling day, show my Idaho drivers license, fill out the registration form, and vote.
The only proof of US citizenship required is that I would tick a box that says that I'm a US citizen.
I can very easily see that being fraudulently exploited, or just plain misunderstood by new immigrants.
I grew up in Australia, where voting is mandatory and has been for nearly a century. The system seems solid, and there is very rarely talk of vote fraud. The history of voting in the US is so full of change and diversity that I'm hardly surprised there are opportunities for exploits like this.