I don't play these, but I've had a couple of tries on a set of uillean pipes, and so I have, perhaps, a little insight into why they are so notoriously hard.
Firstly, the bags. As stated above, one of them (the bellows) is used to draw air into the instrument. Unlike some Scottish bagpipes, which have only one bag, you don't blow into the instrument at all. Air is drawn in by forcibly expanding the right-hand bag, which is kept more or less under the elbow and strapped to the right arm. As you move the arm away from the body, the air rushes into the bag, avoiding the creation of a vacuum. The air passes across the body via a connecting tube to the left hand bag when you move the right arm back towards your torso. The left hand bag will tend to expand as you do this, though you have to maintain pressure at the same time with your left arm. As the right arm goes back out again to suck in more air, the left arm is pressed back in, maintaining the same even pressure, which expels the air through the drones and the chanter. Thus, there's a cyclic motion of the arms you can acquire, which maintains a steady and even flow of air out through the pipes.
This is hard enough in itself, but additionally it's vital to keep a an absolutely constant pressure in the air flowing out of the left bag, because the pitch of the note obtained is extremely sensitive to the pressure of the air against the reeds in the pipes. This is as if the strength with which you bowed a note on the violin, or plucked a string on a guitar, could vary the frequency of the note obtained by as much as a tone or more. One flaw in the cycle of arm movements, allowing the left bag to completely empty, and your tune will wander off into Hindemith-like atonality as you bring the bag back up to pressure.
Skillful (virtuosic) players, like Paddy Keenan, piper for Planxty, Patrick Street, The Bothy Band, and Moving Hearts, can use this effect to 'bend' notes, like Hendrix bends his guitar strings.
Another difficulty is the fingering itself. Unlike the simple fingering scheme on a penny whistle, where you generally finger all the holes down to the first open one, none after, on the uilleans there is generally only one hole on the chanter left unfingered at a time, which can make for some very complicated movements of the fingers, which must often be executed with extreme rapidity, owing to the highly ornamented forms of music usually played on the instrument. (Even in slow airs and laments, rapid ornaments are often required.)
When you hear a master like Keenan execute a fast reel like The Kid on the Mountain, with its many long rolls, and grace notes, using variations in pressure to subtly alter the basic pitch, or playing the jazz-like improvisatory solos on Moving Hearts tunes, making a more dramatic, but still extremely precise, use of the pitch-bending, the effect on someone who's tried their hand at the instrument, no matter how briefly, is nothing less than jaw-dropping.