Voltage to segments on 7 segment display

Actually this is a very common arrangement. Each digit has its anodes connected together, and the segments for all digits have their cathodes connected. The controller enables one of the common anodes, drives the segments, and waits, then turns off the cathodes and moves on to the next common anode. This arrangement and control method is called multiplexing, and the linked article will give you a lot of information. The part in question is a common anode display, but common cathode displays are just as common.

Multiplexing allows you to drive a large number of display elements with fewer wires than individual addressing, at the expense of a more complex control algorithm. You can go a step further by doing something called Charlieplexing which allows you to drive even more elements with less I/O if you can turn off the drivers (as opposed to just setting them high or low, which is the requirement for a multiplexed display controller).

Now that you know it's called a multiplexed display, you should have no trouble finding all kinds of example circuits or microcontroller code fragments to implement it.

The purpose of the resistor is to limit current to a segment LED, by dropping the surplus voltage (over the LED Vf) across itself.

When a resistor is used on a common cathode, the current through this resistor would vary by the number of segments lit at any time. Thus, the voltage dropped would change as well. This will lead to intensity of digits changing with number of lit segments.