Why does a 74HC595N work in LED matrices?
All the LED current flows through the Vcc pin of the 74HC595 (and the GND pin of the TIPxxx). The absolute maximum rated current through Vcc (or GND) of a typical 74HC595 is indeed 70mA. 

The absolute maximum peak current per LED is thus 8.75mA, or an average current of about 1mA per LED (1/8 duty cycle per LED). 

In practice you should stay WELL away from the absolute maximum value. 

To put it explicitly, this is a hobbyist-level circuit, designed by someone who doesn't care or doesn't know much about reliability (assuming they actually recommended anything like the currents you stated). Using the 74HC595 to drive a high-side driver array or prebiased PNP transistor duals would be much better. They're designed as logic shift registers, not as load drivers. 

Using such drivers you could also get a much higher brightness. An average current of 10mA per LED requires a total current of 640mA, obviously, which means that the source drivers need to handle 80mA each (with all potentially on at once) and the sink drivers need to hand 640mA each (with each one seeing a 1/8 duty cycle). 

Edit: You can get a good idea of what kind of average current you want by testing a single LED of the matrix through a resistor. If 500uA or 750uA is enough (and it may well be if you have an optical filter and subdued lighting and high-brightness LED dice in the display) then you can use the original circuit. If you need high brightness (eg. daylight visibility) then you probably need to drive the LEDs near their limits.