Each phasor represents the light coming from a small section of the width of the slit. The length of the phasor corresponds to the amplitude of the light, and the direction corresponds to the phase. Adding waves together corresponds to adding the phasors by putting them head to tail daisy-chain style and drawing the resultant phasor directly from one end of the chain to the other. The length of the resultant gives you the amplitude of the sum of the waves.
You divide the width of the slit into a lot of sections (call the number N) all with the same width. The light coming from each section corresponds to a phasor, all with the same length, but with slightly different directions. The angle between phasors depends on which point on the screen the light arrives at.
At the center of the screen, all the light is in phase, so the phasors are all in the same direction and they add in a straight line. The amplitude (length) of the resultant is just the sum of the amplitudes of the individual phasors.
Move a little bit away from the center of the screen, and the phasors are now in slightly different directions, with consecutive phasors having a small angle between them, the same angle for each pair. The daisy-chain starts to curl away from a straight line.
As you move further from the center of the screen, the chain of phasors curls up more and more, but the total length along the chain remains constant. Think of curling up a piece of string with a fixed length. The straight-line distance between the two ends decreases as the chain curls up. Eventually the two ends meet, forming a circle, and the resultant is zero: the first minimum of the diffraction pattern.
Now keep going further from the center of the screen. The chain of phasors continues to curl up on itself, with the two ends overlapping. They still form a circle, but the radius decreases as the the ends overlap more and more. The distance between the ends (the amplitude of the resultant) increases from zero at first, reaches a second maximum near where the chain makes "one and a half" circles, and then becomes zero again when the chain makes two full circles. Etc. The maxima get smaller and smaller because the diameter of the circle gets smaller and smaller.
In the derivation, δ (delta) is the total phase difference "across the width of the slit", that is, between rays of light from the two edges of the slit, represented by the first and last phasors in the chain. At the first minimum (phasor chain makes one complete circle), δ = 360° (2π radians). At the second minimum, δ = 720° (4π radians), etc.
