I know this thread is a few weeks old, but it has a lot of responses, and my name was mentioned as a source. For those reasons, I would like to expand on some of the information presented here, and hopefully clear up a few misconceptions.
First off, some terminology.
The router spindle (orange/brown) is part of the motor shaft, and is hollowed out to contain the internal taper to mate with the collet. It is threaded on the end to secure the collet nut. The spindle depth is the limiting factor as to how deep a router bit can be inserted, especially a straight fluted bit, as shown.
The collet (green) has an external taper matching the internal taper of the router spindle. As the collet gets pressed deeper into the spindle, the mating tapers cause the collet to become compressed--the internal diameter of the collet gets smaller, and this grips the side of the router bit.
The collet nut (light blue) presses the collet into the spindle. Most modern routers have a captive collet nut, such that the nut and collet are locked together. The benefit of this, is that as the nut is unscrewed, it actively "pulls" the collet out of the spindle. Prior to the development of captive collet nuts several decades ago, the collet nut could be completely unscrewed, and the collet would still be stuck inside the taper due to friction.
History:
Years ago, Hitachi came out with the M12V router. It had a very shallow spindle bore, and the collet itself was threaded (no collet nut). The spindle bore was so shallow that many/most router bits could touch the bottom of the spindle bore. This prevented the collet from being properly tightened.
The Hitachi M12V router was one of the most popular routers of its time. Because so many M12V owners were talking about this problem of bits bottoming out, many other woodworkers assumed it was a common problem for all routers. In reality, most routers had deep enough spindles that it was nearly impossible to bottom out a bit in the spindle.
Misinformation Side Effect:
Because the topic of router bit insertion was coming up so frequently, the topic ended up morphing into the idea that all router bits needed to be inserted to the point of bottoming out, and then retracted slightly...even for routers without this shallow bore.
This led to a new version of the topic where woodworkers believed that router bits needed to be inserted almost all the way until the head of the bit reached the face of the collet. Not only did this result in a lot of carbide getting chipped during removal, but also ended up with router bits not getting tight sue to the fillet between the shank and head of the router bit.
Proper Bit Insertion:
Ignoring any reasons to do otherwise, the proper insertion of a router bit for maximum gripping of the bit by the collet, is to insert the bit to the same depth as the "Gripping Surface" (dark blue) of the collet. This is shown in the right-hand image.
Inserting the bit deeper than this will not increase the grip.
Inserting the bit less than this will decrease the grip.
Exceptions to Bit Insertion:
As I prefaced above, "Ignoring any reasons to do otherwise"; here are some of those reasons.
If you have a router bit with a very large diameter head, especially if it has a small shank (1/4 inch), then you may wish to insert the bit deeper so less of the router bit's shank is exposed. The same is true if you have a router bit with an unusually long shank.You are not doing this for greater grip on the bit, but instead, to reduce vibration and chatter in the head of the bit.
Exceptions for inserting a router bit shallow should be taken with care. Too shallow, and the grip can be dangerously unstable. However, it should be noted that every router manufacturer uses their own collet designs, which have different lengths of gripping surface. The graphics shown below are exact representations of the Festool OF1400/OF2200 collets and nuts. These collets have approximately 1-inch of gripping surface. If the need arose, and the router bit was small, you could reduce the insertion depth slightly, simply because this collet has such a long gripping surface.