It's a little more complex that this. Let me try and explain it again. :grinning:
With the matrix, you need to switch a cathode LOW and an anode HIGH in order to light a LED. But as MarkT points out, LEDs require a current-limiting resistor.
Since we will want to (presumably) have eight rows and eight columns in the matrix, let's for a moment presume you have a matrix in which cathodes are column terminals and anodes are row terminals. This is the "B" version of the common matrices.
Since we have eight rows, we will put one resistor in series with each row (anode) wire. Given that the LEDs are red, they will have a voltage drop of around 2 V. The Arduino drivers will lose about half a Volt at 20 mA and you have both an anode and cathode driver so you lose a volt between and have two Volts remaining across the resistor, so for 20 mA you use a 100 Ohm resistor.
This would be satisfactory if you only lit one LED at a time, but in practice, you usually want to light anything up to eight LEDs in the same column at once, so the column would then want to draw 160 mA which is not allowable (or even possible) for an Arduino pin. So you actually need to make the resistors 1k2, limiting the current for all eight LEDs in a column to 16 mA - 2 mA for each LED.
But now the LEDs will not be so bright, and each column only gets one eighth of the time to be lit as you multiplex it.
If you actually want a nice bright display, instead of driving the matrix with an Arduino, you need to use the purpose-designed MAX7219 chip which controls the LED currents using a single "programming" resistor and performs all the multiplexing for you; you just write the patterns (in eight bytes) into its registers.
And these are (or were inexpensively before Covid-19) available as modules including the LED matrix itself on eBay or Aliexpress. While you can experiment with just the matrices or a quantity of LEDs, resistors and the Arduino, these modules are the way to go when you want to use it for some purpose. You can use the modules which come unassembled, ignoring the matrix provided, if you want to assemble your own matrix of LEDs or to control a large number of LEDs spread out over a piece of equipment, model or some other sort of display.