Severe disability premium is all about increasing your entitlement to other, means tested benefits. Essentially a means tested benefit like Income Support or Housing Benefit is based on a calculation, comparing what income you have coming in each week against what the government says you need to live on. (And yes, this is peanuts!)
When you qualify for enhanced rate of PIP for Care, you may qualify for the severe disability premium. This means that the amount of income you have can be higher because your disability indicates that the amount against which your income is compared is higher.
It’s not precisely something that you need to apply for because it’s not a benefit in its own right. It just means that if you already get another benefit that is means tested, you should tell the awarding office (the council or JSA office) that you get PIP at the Enhanced rate. And obviously, show them proof.