d<λ - avoiding phase wrap around
d>n*λ - new grating lobe emerges for each n*λ (phase wrap around)
After d>λ relative element phases start to wrap around through 180° and 360°, giving zeroes and peaks.
It all comes from summation, try to consider trigonometry regardless antenna design.
You have N inputs of same frequency tone with certain phase offset.
If you want to understand how radiation is added intuitively, I think it is better to use more "low level" approach without using all "simplifications" from antenna theory books.
Assume we have patch at position (x1,y1,z1) and some point in space where we measure field (x2,y2,z2). All coordinates are normalized to wavelength λ. Distance between these points is d1(x1,y1,z1,x2,y2,z2)=Sqrt( (x2-x1)^2+(y2-y1)^2+(z2-z1)^2)
Then signal from each antenna signal with phase shift in exponential form for all those distances may be written as sig[N]=Exp(i*d1(x1[N],y1[N],z1[N],x2,y2,z2)*2*Pi)
And after summation of all sig[N] for N patches you can calculate 20*Log10(Abs(...)) of this sum to get radiation pattern value for point x2,y2,z2.
Here is another example (with less math). Let's call it "two points example":
Imagine first point in front of antenna at distance of 100 meters. For all patches distances to this point is almost the same, so phase shift will be almost the same (360*distance/λ=360*100/λ), which provides almost 0° relative phase shift (constructive addition)
Imagine second point, and put it on the left relative to antenna, at a distance of 100 meters.
If elements spacing is 0.5*λ then relative phase between adjacent patches will be 100+0.5*λ, 100+0.5*λ+0.5*λ, 100+0.5*λ+0.5*λ+0.5*λ and so on. 100 meters distance is the same for all patches, we only need relative offsets, which are 0.5*λ, 0.5*λ+0.5*λ, 0.5*λ+0.5*λ+0.5*λ.
If you imagine point smoothly moving from first position to second position (through a quarter-circle trajectory), then you will see how relative phase shift increases over angle.
point1: 0, 0, 0 (in front of antenna)
point2: 0.5*λ, 0.5*λ+0.5*λ, 0.5*λ+0.5*λ+0.5*λ (on the left side, rotated 90deg relative to broadside)
If d=1*λ point2 will have grating lobe, but element factor of radiator improves situation, because radiation is very weak at this direction:
point2: 1*λ, 1*λ+1*λ, 1*λ+1*λ+1*λ (constructive addition for the point on the left, the same as at broadside)
if d>λ, then point2: 1*λ, 1*λ+1*λ, 1*λ+1*λ+1*λ situation will emerge earlier, at some smaller angle, and start to be a problem.
Depending on element spacing d constructive addition may occur at several angles, not only at 90°.
If somebody read this later, please note that some sources use 90° (angle changes from 0° to 180°) for broadside and other use 0° for broadside (angle changes from -90° to 90°). And formula for spacing*cos(ang) will use sin(ang) if angle is measured relative to broadside axis.