First, imagine your input space as being in the first quadrant, where there's a unit square with vertices $(0,0), (1,0), (1,1), (0,1)$. Draw this out. Now draw the line $y=2x$. If you pick any point $(x,y)$ where $x \in [0,1], y \in [0,1]$, it will fall somewhere in this unit square. You need to figure out what part in this square satisfies $y < 2x$. I find it easier to work with the probability that $y \geq 2x$ and then you can easily find the other probability. So if we're trying to find the probability that $y \geq 2x$, you shade the upper part of the triangle bounded by the line $y = 2x$ and the top of the unit square ($y = 1$). Color this triangle in.
You can now compute where the line $y=2x$ intersects the line $y=1$. From here, you now have the width and height of this colored-in triangle. Compute the area of that, and divide by the area of the unit square. This gives you the probability, which comes out to be $.25$. This is the probability, remember, that $y \geq 2x$. To find the complement of this probability, that $y < 2x$, you just do $1 - .25 = .75$.