Why doesn't ua741cn operate correctly with a single supply?
It doesn't need a dual supply.  It just needs to operate in a space that has some distance between the negative and positive rails, and for most common power supply schemes, that space is pretty small if the negative rail is at ground, and the arrangement isn't very useful.

As to "why" -- it's the configuration of the transistors in the device.  I'm not sure you're interested in the gory details, and I haven't checked them, but often this has a lot to do with bias voltages used to keep transistor amplifiers in range for linear operation.  When the chip was designed, just having access to a device as facile as an op amp was a pretty good thing.  These days, we want less batteries with more life, less power, smaller voltages, and an op amp designed so long ago just doesn't fill the niche anymore.

If you look at the datasheet at http://www.ti.com/general/docs/lit/getliterature.tsp?genericPartNumber=UA741&amp;fileType=pdf, you'll see in section 7.3, the \$V_{ICR}\$ line, that the common mode input needs to be about 3 volts above the negative rail and 3 volts below the positive rail.  The \$V_{OM}\$ line shows a similar range for the output.  Note that this means the chip is entirely unsuitable for power rails at 0V and 5V, but there is a bit of room if you're operating around 5V for rails at ground and 10V.

So, the "why" isn't there, but the "will my op amp work the way I want it to"  is right there for anyone who knows how to find and read the instructions.