Why does the Electric Field of a dipole have no $x$ component? 
According to the solution $2E\sin(\theta) = E_{net}$.
I understand how the dipole works but I don't understand why the x component would be 0? 
 A: The sum of the x-components (and z-components) is zero at any point on this line you have on your picture, which is exactly in between the two charges. 
Because here the point is exactly at the same distance and direction from each charge. This means that both charges apply the same magnitude of electric field; but since they have opposite charges, the field directions are opposite (one charge "pulls" inwards with that magnitude, the other "pushes" outwards with that magnitude). The direction of the axis is the same for the two charges, so it cancels out.
Equal distance is also true for the y-components of course; the magnitudes of the field applied by each charge are equal, but here the positive axis-direction is not the same for both charges. If positive is the straight way from minus-to-minus, then the negative charge pulls and the positive pushes, both in the same direction. So here instead, the fields add up.
See the drawing below. Arrows that point straight vertically up or downwards do not have any x-component of electric field.

A: Specifically, the electric field of a dipole has no x-component ($E_x=0$) at points along the specific axis you have drawn. The points along this axis are equidistant from both the charges, so the electric field produced by each charge is equal in magnitude, but different in direction since the charges have opposite sign and are located in different places.
Now, from a symmetry point of view, if the charges were exactly on top of each other ($d->0$) then the electric field would be zero everywhere since the two fields would exactly cancel. However, since the charges are vertically displaced relative to these points, this displacement corresponds to the generation of non-zero vertical electric fields along this axis. They are not displaced horizontally relative to these points, so the electric field in the horizontal direction remains zero. In your figure, "vertical" corresponds to the y-direction and "horizontal" corresponds to the x-direction.
If you are still having trouble seeing it, try drawing the vector lines produced by each independent charge.
A: The diagram shows, alone the vertical lines (in between q and -q) all fields are in the 
z direction (horizontal, according to the diagram). at all other points, the field has components in both the vertical and horizontal directions.
