Why should the leading edge be blunt on low-speed, subsonic airfoils?
When using an airfoil with a sharp leading edge in subsonic regime, you need to adapt the angle of attack so that the stagnation point occurs right on the sharp edge.

At every flight condition there should be an AoA which achieves this. Higher AoA, and the stagnation point is too low. To low AoA, and the stagnation point is too high. However flight conditions change frequently, so the required AoA changes dynamically as well. Since AoA also drives the lift and lift is the priority in a pilot's mind (he wants to keep flying after all), stagnation point will only occasionally match the leading edge.

During flight the AoA will often fail to match the stagnation point, let's say with too high AoA, separation would occur below the leading edge (on the "intrados").  This is the only good illustration I could find:



The stagnation point is at the tip of the only grey line that stops right on the black plate.
Basically all incoming flow between these leading edge and stagnation point  (unfortunately no line shows this on the picture) will follow this trajectory:


Incoming at high speed onto the airfoil, below leading edge, going towards the stagnation point, but aims just slightly higher up.
The stagnation point is the point of highest pressure, so the incoming flow slows down
Comes a point where the slowdown is sufficient to actually reverse the flow which now shoots up along the surface towards the leading edge, now driven by the pressure from stagnation point
Arriving at the leading edge from below the flow needs to do a U-turn around the leading edge to keep following the wing shape and flow towards the "extrados" and keep on with its life.


Therein lies the problem: when finally arriving at the leading edge, the air has non-zero speed (zero speed only occurs at stagnation point, of which there is only one), but must instantly reverse it. Physics says nope, and you get a large leading edge vortex to smooth the motion out. Unfortunately that is also called a flow separation bubble, which may became a stall.

This is:


bad for performance (drag)
bad for safety (stallable)
bad for comfort (vibration from the eddies) 
bad for structure (vibration fatigue)


Why does it only happen in subsonic flight?

At supersonic speeds, information cannot travel back on the flow using the same pressure gradient trick, once the air is on the intrados/extrados, it stays there, and cannot back up the other way. So no 'adaptation' is possible, regardless of AoA.

Why use rounded leading edge to remedy this in subsonic?

With a rounded edge, the U-turn is not so sharp, it is actually manageable for the air. This means you'll not always have flow separation, so you're safe. The rounder, the easier larger the range of AoA stay "adapted". This animated gif shows really well how the stagnation point can move around the rounded tip thus avoiding separation:



Source: UNIVERSITY OF GENOA
FACULTY OF ENGINEERING
DEPARTMENT OF ENVIRONMENTAL ENGINEERING
IRROTATIONAL PLANE FLOWS OF AN INVISCID FLUID

TL;DR It's possible to fly subsonic with sharp leading edge, but less safe and lower performance. The key is AoA adaptation.