In the double slit experiment, why is the distance between the slits and the screen proportional to the fringe separation? I know it is true by the formula, however I would like a qualitative description please. 
 A: I think pictures provide the best qualitative descriptions.
I used matlab to plot the interference pattern  caused by two sources placed at a certain distance and tried to underline where the fringe will appear on screens placed at different distances.
Let's first recall that the distance $t$ between the fringes is given approximately by:
$$
t= \frac{D\lambda}{d}, 
$$
where $D$ is the distance from the screen, $\lambda$ the wavelenght, $d$ the distance between the slits.
 
On the left you can see the interference pattern, while on the right the red lines represent the paths of the crests of the wave where constructive interference is happening. The fringes appear where the red lines cross the black ones, representing two screens.
You can see that the distance between the fringes increases linearly with screen distance from the two slits.
In the next plot I doubled the distance between the two slits: the previous formula predicts that the distance between the fringes will halve. And is what happens, indeed.

A: When you join the points of space having a constructive interference, you form hyperbola (in the classical case). These curves are close to one another near the slits and get more spaced near the screen. 
A: The points of the bright maxima are given by the expression:
$$ d\theta = n\lambda, \ \ \ n = 0, 1, 2, ...$$
Where $\lambda$ is the wavelength of the light. Then the angular spacing of the fringes is given by:
$$ \theta_f \approx \lambda/d $$
Where d is the separation of the slits. But since this is the angular separation, we must account for the distance to the screen to get the actual distance between slits as measured on the screen. So, taking this into account, let $D$ be the distance to the screen, then the separation as measured on the screen is:
$$ w = D\theta_f \approx \frac{D\lambda}{d}$$
