Intuition behind the positive slope of LM curve : 'Income must increase'?
The intertemporal consumption equation is :

$$Y_{1}+\frac{Y_{2}}{(1+r)} = W$$

where $W$ is total wealth and $Y$ is revenue.

When interest rate rise investiment drops and savings rise, so lets say someone choses to save all his revenu from period 1 (this is just to calculate his total wealth), then in period 2 his total wealth will be :

$$Y_{1}(1+r)+Y_{2} = W(1+r)$$

So as you can see his total wealth in 2nd period grows when interest rate grows 

Also we can isolate $Y_{1}$ : 

$$Y_{1}= W-\frac{Y_{2}}{(1+r)}$$

And you see that when interest rate rises your revenu in period one rises.

I am not sure if this is a valid answer but it seems correct