I'm a bit confused by why it says "report a gain when interest rates increases" for bonds payable. I thought that if rates increase the size of your liability increases as you need to pay back more...

Bond prices move in an inverse relationship to interest rates. As rates rise bond prices (liabilities) fall. So if rates rise the liability falls and a gain is reported in the income statement. The yield represents the discount rate used to work out the PV of the bond an incorporates the risk of investing in the bond.

PV = FV / (1+r)^t, if r goes up the PV of the liability goes down.