It sounds more complicated than it is.
There are two possible outcomes when you first pick - you pick the winner (1/3), or you pick the loser (2/3)
If you don’t switch, your chances remain 1/3 because nothing has changed.
If you do switch, the only way you can lose is if you originally picked the winner, since otherwise the host has just removed the only remaining losing pick. Your chances of picking a loser originally are 2/3, so your chance of winning if you switch is 2/3.
The confusion/peril comes from the worry about picking the right one and switching away, which would feel bad.
I realise no-one asked for it to actually be explained, but I do love probability, so, 