Why does my lamp fixture say either 60 watt incandescent or 15 watt CFL MAX?

The old rule of 60w maximum was ostensibly about the fixture's ability to stand the heat generated by incandescent lamps. They do their thing by getting literally white hot, after all.

Really, the rule has everything to do with a company wanting to shed as much liability as possible. If you go and burn your place down because you used a 100w bulb in a fixture rated for only 60w, then it's all your fault, right? Right. Because you ignored a safety warning.

This new rule is because somebody didn't understand the rationale behind the old one. Heat from the incandescent lamp was the problem there and always was. CFLs get warm but nowhere near as hot as an incandescent lamp. You can touch a 100w-equivalent CFL while it's at full brightness and has been for a while; it'll be uncomfortably warm but you won't be burned like you would on even a 40w incandescent. But all the ratings on a CFL give an incandescent wattage equivalency, and that's what the rule makers are looking at. They aren't thinking, that's all.

You could safely operate a 60w CFL in a 60w-rated incandescent fixture - assuming you could get the lamp to fit. It would give off light comparable to a 250-300w incandescent lamp. It would be enormous compared to the incandescent lamp. But the dissipated energy as heat would be the same, and so the fixture would be able to handle it.

The short answer to your question is the the 15W CFL is the equivalent in output to the 60 W incandescent bulb and since CFL do not even run close to the temps incandescent bulbs do You could use say a 18 watt CFL and it'll be okay but if you ran a 100W incandescent you do the risk of damage and even fire because of the very high temps the incandescent will generate .Thats how big a difference in heat the two bulbs generate

A 15 CFL puts out as much light as a 60 watt incandescent

The MORON that wrote the label actually believes that a 15 watt CFL is equivalent to a 60 watt incandescent. While the light output is close, that doesn't actually matter for the fixture. The fixture is designed to handle the current draw of a 60 watt bulb. The only reason you can't safely use a 60 watt CFL is they don't make them.

LED is the best, they now make them with plastic bulbs so they are much more robust than cfl types which use thin glass. A 10 watt LED lamp will give as much light as a 100 watt filament, but produces only 2 watts of heat. A 60watt filament produces almost 50 watts of heat.

Even better, just switch to an LED bulb and stop worrying. They do not generate heat and use only a fraction of the electricity.

the CFL burns almost cold and will last longer - the CFL is brighter at lower wattage .....

I'll put my faith in God. NOT the religion and culture which has taught you to hate.