The reason why the twins will think that their clock is running faster or slower than the other twin's is, is because they are both unable to observe the other twin "from above" (like we do in the picture that you provided). They instead, observe each other while considering themselves as stationary!
But that's "wrong"...* Neither of them is stationary... They are both moving in spacetime. But since we have the equivelence principle, we can infact consider ourselves as stationary and think that they other guy is moving in space or in time, either faster or slower... That creates the "paradox" that you are asking about. Its not a paradox, is just shortsightedness...
(Its not really wrong, we are just unable to see spacetime "from above" like in the picture you provided, so we end up seeing all those effects of relativity, like time dilation and length contraction. When we observe both objects from "above" all those effects vanish).