Sure.
> Why do you refinance if rates go up? Surely the point is that if rates go up you've locked in a better rate
You don't have to, but you can choose to either (a) keep the same rate and owe the same amount, or (b) get the new (higher) rate and owe less.
> How does half your debt disappear if rates go up?
It doesn't exactly. However, the market value of your mortgage loan halves if the rate doubles (roughly). This means you can:
1. Take out a loan for half the original amount, at double the interest rate
2. Buy back your original loan
3. Destroy the original loan (which is fine since you are both creditor and debtor for that loan)
This leaves you with the loan taken in step 1.
To understand why it works this way it's instructive to think of a loan as an exchange of wealth for income. One party has some savings (wealth) and would like to exchange it for an income (e.g. to pay recurring expenses). Another party has an income and would like to trade it for wealth (e.g. to buy a house). Viewing a loan in this way, the interest rate is nothing more than the current price of a certain income stream (measured in percent per year).
For example, let's assume that the current market price for an income of $2 per year is $100. This is equivalent to an interest rate of 2% per year. I have $100 that I'm willing to part with for an income of $2 per year, and you have an income of at least $2 per year that you're willing to sell for $100. We make the deal. Now, the day after we shake hands to make the deal, the current market price for an income of $2 per year falls to $50. This is equivalent to a doubling of the interest rate (from 2% per year to 4% per year). I still have the income of $2 per year that I paid $100 for yesterday, but if I want to sell this to someone else I can only get $50 for it. And if you were to ask me to buy back the loan for $50 I would have nothing against that, since I could go out immediately after and buy the same $2/year income for those $50.