Here is a geometric explanation. See if it makes sense to you.
$(X,Y)$ is a uniform random point in the square $[0,1] \times [0,1]$.
But what is $(W,Z) \equiv (\min(X,Y), \max(X,Y))$?
Imagine folding the square along the $y=x$ line, aka $+45°$ line, folding the lower triangle onto the upper triangle. Then $(W,Z)$ is where $(X,Y)$ would end up:
If $Y>X$, the original point is in the upper triangle to begin with, and remains there (folding has no effect). In this case $(W,Z) = (X,Y)$.
If $X > Y$, the original point is in the lower triangle to begin with, and ends up in the upper triangle at $(Y,X)$ (that's the effect of folding). In this case $(W,Z) = (Y,X)$.
By symmetry, therefore, sampling $(X,Y)$ uniformly in the square and then computing $W=\min(X,Y)$ and $Z=\max(X,Y)$ is equivalent to sampling $(W,Z)$ uniformly from just the upper triangle.
I hope it is now obvious, because a triangle is not a square, that $W, Z$ will be dependent.
As for the sign of the correlation: Very roughly speaking positive correlation means the area is "biased / slanted with a positive slope" and negative correlation means the area is "biased / slanted with a negative slope". I hope it is also obvious that the upper triangle implies a positive correlation.
(Incidentally, as you probably know, dependence can still mean zero correlation, and some examples would be uniformly sampling from e.g. a circle, a $45°$ rotated square, etc.)
Sorry the geometric argument is kinda vague, but I thought this kind of "intuitive" picture is what your question is asking (since you already know the answer is $1/36$ and just need an explanation which is not the actual calculation of covariance). If you want a slightly less "visual" reason, simply note that, conditioned on $W=w, Z \sim Uniform(w,1)$, so as $w$ increases $Z$ certainly tends to also increase.
Re: your multiple samples argument: that doesn't quite apply. If you sample many $(W,Z)$ in the upper triangle, then sure $\min_j W_j \rightarrow 0$ and $\max_j Z_j \rightarrow 1$, but so what? That doesn't say anything about how in one particular sample, knowing $W$ gives you info about $Z$ and vice versa. (In fact, just for fun, it is also true that $\max_j W_j \rightarrow 1$ and $\min_j Z_j \rightarrow 0$, so what do you make of that? ;) )