The interest rate itself falls

Demand for loanable funds stays put. Supply of loanable funds stays put.

The real interest rate is the variable on the vertical axis. A change in it moves borrowers and savers along their existing curves. At a lower rate, borrowers want more funds and savers offer fewer, both movements along the curves. Neither curve shifts. If something had shifted, the stem would have said what.

The quantity of funds demanded rises and the quantity supplied falls along the unchanged curves, and neither curve moves.

Common mistake. The usual slip is to shift demand right to 'show more borrowing.' More borrowing at a lower rate is a movement along the demand curve. Only changes in the reasons to borrow or save shift the curves.