In the loanable funds market, the main reason we believe that the supply of loanable funds slopes upward, i.e. more funds are offered for borrowing if the real interest rate is higher, is that

1. loanable funds are mobile internationally

This is the right answer. Remember, our model is always ceteris paribus, i.e. everything else unchanged. So, in the supply and demand diagram for US loanable funds, we measure the US real interest rate on the vertical axis, the US quantity of loanable funds on the horizontal axis, for the US market. When we move up on the vertical axis, increase the US real interest rate, the real interest rate in London, Frankfurt, Tokyo, etc is assumed unchanged. Short term capital is highly mobile, so the increased real interest rate in the US will cause loanable funds to switch from those foreign markets to the US market, increasing the quantity of loanable funds offered in the US market -- so the supply of loanable funds slopes upward.
 

2. households are bound to save more if real interest rates go up

Much as many people would like to believe this, we cannot be sure of it. There is both a substitution effect -- the higher real interest rate means the opportunity cost (interest foregone) of consuming now is higher, so we would expect households to save more, consume less -- and an income/wealth effect -- the higher interest rate means real lifetime income is higher, we should tend to consume more, save less [suppose you were trying to save enough to give an income to retire on; with the higher interest rate you now need to save less to achieve the same target]. The two effects go in opposite directions and we have no way of knowing which will be bigger overall, so this is not a convincing reason why the supply curve should slope up.
 

3. if the real interest rate is higher, it is more expensive to borrow

This has to do with the demand for loanable funds, not the supply.
 

4. the real balance effect