There’s no real scientific explanation behind why some people experience itchiness when they start keto. There’s just a handful of occurrences that people have written about, so I’m basing my answer on what I’ve read.
From anecdotes, it’s most likely irritation from the acetone that is excreted in sweat (it’s why you may experience bad breath).
It’s worth looking into better clothing options for absorbing or wicking sweat from your body. It’s also worth showering right after activity that causes you to sweat.
If it’s a lasting issue that is causing problems, you may want to consider upping your carbs or changing exercise plans.