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, seemingly insignificant speck can wedge a shank in off center. It won't take very much dirt or dust at all to interfere enough to cause vibration and slipping. If the crumb disintegrates, the bit may unexpectedly slip loose. 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, repeated loosening and tightening bends the grippers. Metal fatigue gradually stiffens the collet until it gets reluctant to release. Eventually, 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 one 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. That will only force it to get tighter. Unscrew it in the same way it spins. Some newer models may seem to be seized up, but it is only temporary until their second stage begins releasing. To those accustomed to older models, this belated loosening may be a surprise and relief.
Forget about banging a hammer to dislodge a frozen bit. Don't seize it with pliers to 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 hold it still with a wrench. Wrap a thick rag around the stuck bit, then twist it. If you still cannot extract it, or if it just won't budge, then you may need to unthread the collet so you can fit a punch in the bottom to gently tap on the end to remove the bit.
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, twist a brass brush inside it with alcohol, 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. Torque should be moderate and consistent each time you change a bit. It may get tighter in use.