Lets look at a couple of cases.
1. Both curves are above the x-axis
Minusing the area under the bottom curve from the area under the top curve give just the orange area, the area between the two curves.
2. One above and one below the x-axis
The signed area above the axis is postive and the signed area below the axis is negative. We minus the area of the bottom curve, that is we minus the negative, this results in the addition of the area between the axis and the bottom curve to the top curve. That is yellow - blue (blue is negative) so the areas are added.
3. Both below the x-axis
The area between the curves and the axis are both negative, but the bottom curve's area has a greater negative. When we take this greater negative away, we add a greater value and so end up with a postive result for the area that is between the two curves
4. Partly above and partly below
Here for the sections of the top curve that are above the axis, we are just doing case 2, minusing the negative so that it adds. For the part of the top graph that is under the axis, that is the green part under the axis, we can explain it with case 3. The greater negative is minused to give the postive value, the area between the two curves.
I hope this makes sense and I haven't confused you more.