What does "Run Power and Ground wires back to the Arduino" mean?
Source: Adafruit Learning Systems, RGB Strips
This is the PDF version of the article.    






  Is this a typo on their end or is there something I'm missing here?


Not a typo.  Not sure it deserves two highlighted warnings.  

Here is a schematic of the UNO power distribution:




Source: elecrom.com





The UNO gets its 5V from the output of an on board 5V LDO.
This LDO input is Vin
The LDO drops the 12V down to 5V.

I assume "back to the Arduino" means to the PWRIN or VIN connector.

When the current to the strip is over an amp, the wire can drop a significant amount of the voltage when long and/or thin wires are used.  

So as shown,  the power supply is connected to the strip with short wires then long wires can be used to power the UNO.  Any drop in voltage from the strip to the UNO is helpful in reducing the 12V and reducing the load on the LDO. 

This also assumes the UNO 5V and 3.3V are not used to power other add ons.