Why do bits get stuck?
A router collet has just enough space in it to accommodate tightening. During use, the metal expands. If any sawdust gets in, it can neither grip nor loosen easily. A tiny speck can wedge a shank in off center. Dirt can interfere enough to cause vibration and slipping. If the crumb disintegrates, the bit may unexpectedly slip loose during use. The quality of cut improves if the parts are fitting together smoothly without interference.
Collet damage
A new collet starts out free of sticking or slipping, but nothing lasts forever. Each time you're changing a bit and tightening, it bends. Gradually, metal fatigue stiffens it, so it gets reluctant to release. It starts to slip and encourages you to repeatedly tighten it too far. As the nose gets dinged and damaged, it spreads like a bell and the portion of it securely gripping shrinks. Replace it if it is deformed and not gripping firmly or releasing reliably. Put the new collet in the nut before threading onto the spindle. A good one has tolerance under .0005 inch. It is springy and has several slits. It holds on center and doesn't vibrate.
Shank condition
If you've been tightening to an extreme, you'll start to see fretting on shanks. If not in use, leaving bits in can cause distortion. As a collet tightens on a burr or any other sort of minor defect, its sides may deform slightly. If a shank is bent or not perfectly round, the grip is inferior. Later as you are changing a bit, it gets frustrating to remove. A bend may not be visible, but a sign is vibration. Bits put in partway may bend. An insubstantial shaft supporting a big head can flex. 1/4 inch shanks get bent easily in contrast to 1/2 inch. The grip on a slim shank is prone to slipping. If it has been taking a pounding, it tends to stick as you are trying to take it off.
Release a stuck bit
First, how not to change a router bit: Do not loosen the nut against the spin; it'll get tighter. Unscrew it in the same way it spins. Some newer models just seem to be frozen before their second stage begins releasing. To those accustomed to older models, as this type belatedly opens, it may be a surprise and relief.
Forget about banging a hammer to dislodge a frozen bit. Don't use pliers to seize and pull it out. Think twice before oiling it, unless you plan on swabbing up and removing every drop once you take off the stuck bit. Any slippery residue could preclude a good grip later on.
Put on gloves. Lock the spindle or use a wrench to hold it still. Wrap a thick rag around the stuck bit, then twist it. If you cannot extract or budge it at all, then you may need to unthread the assembly so you can fit a punch in the bottom to gently tap on the end to remove it.
Install a router bit
Preparation for installing a bit includes inspecting the shank for scratches. Gently smooth it. Meticulously file off burrs, but avoid scratching the rest of it. Wipe off anything sticking to it. To prepare the collet for installation, use a brass brush and alcohol inside it, then air it out.
Insert a bit mostly in, but not quite to the end. Back off and leave some room. Do not try to maximize its reach by inserting it halfway. 90% of the shaft ought to be fitting inside to ensure it doesn't loosen. The grip is at the nose, so don't insert the taper as you install it. Tighten the nut against the spin. Use a consistently moderate torque each time you change a bit. It may get tighter in use.