This isn't an exact description but it should give some insight.
I think it's better to tackle this problem in stages. First what happens if we drop a horizontal paper? Well, it's edges curve up like this:
Now what happens with 45° paper? Its edges curve up again. Pressure drag is a very sensitive function of angle of attack. It's maximum for a plate perpendicular to the flow drops to zero for a plate parallel to the flow (at which point friction drag dominates). It drops rapidly (most empirical relations predict something between $cos(\theta)$ and $cos^2(\theta)$). So center of pressure is shifted to the left as you see in the picture (red arrows represent pressure vectors).
After gaining some horizontal speed the relative airspeed vector changes so that there's even less pressure on the right edge (shifting the center of pressure even more) and it drops faster than the left edge while the left edge curve up more until it stalls and the situation is reversed. It means that it is unstable, like the second plane bellow:
Now for a positive lift airfoil to be stable its center of mass should be closer to the leading edge than the center of pressure. That's what the weight does. It keeps angle of attack of leading edge low so that it does not stall.
Update That's why I said this isn't exact description. 441
When an object is placed into a flowing fluid, some of it will flow over the object, some of it under. This is a wind tunnel test of a flat plate.
The dividing line is termed stagnation streamline and it's terminated at stagnation point on the object. The fluid will be stationary at this point. As you see in the picture for low angle of attack it's very close to the leading edge.
According to <PERSON>'s principle, in a laminar flow pressure will be greatest where the fluid has no velocity (i.e stagnation point) because at that point all of the dynamic pressure of the fluid will be exerted at the object. So at the lower surface pressure will be greatest at stagnation point, but what about upper surface? Velocity distribution and consequently pressure distribution is not uniform at the upper surface but they're also in favor of leading edge. This figure belongs to the Aerodynamics book by <PERSON> (and it explains in chapter IV how they arrive at this):
Don't look at the left one, that's for supersonic. So center of pressure is closer to the leading edge for subsonic speeds at low angle of attack. But for very high angle of attack stagnation point is close to the middle and pressure distribution is uniform at the other side so center of pressure is at the middle. It's merely an empirical observation that for low angles of attack center of pressure is about quarter of the chord length behind leading edge. 568