The LM curve identifies combinations of income and the interest rate for which the demand for money equals the money supply. In an interest rate/income diagram this curve slopes upward for the following reason:

Pick any point on the LM curve and call it point A. Since the point is on the LM curve, the given money supply is exactly demanded at the indicated interest rate and income level. Now suppose income rises, which means that we are moving horizontally to the right, say into point C. Since people spend more at higher income levels, they wish to hold more money too.

Hence in C the demand for money exceeds the unchanged money supply. To drive the demand for money back down to the initial money supply, holding money needs to become more expensive, that is the interest rate must rise until we are in point B. Hence the money supply slopes upward, meaning that if income increases the interest rate must rise as well to keep the money market in equilibrium.