These matrices are fun to play around with, but their size makes them difficult to use on most breadboards. Here's one way to mount it on a small breadboard in a (relatively) neat way.
1 8x8 LED matrix (this one is 1588BS)
Standard yellow, blue, grey, and brown jumper wires. (4 of each)
(Optional): 4 additional yellow wires OR 4 green wires.
Step 1: Yellow
Connect yellow jumpers across b8-b12, i8-i12, b19-b23, and i19-i23.
Step 2: Blue
Connect the blue jumpers next to the yellow, as shown, with the blue wires on the inside, connecting one pin above and below the yellow.
Step 3: The Rest
Repeat the process with the remaining colored wires.
Step 4: Connect
Connect the matrix to the breadboard so that the top pin on the left connects at a12.
Step 5: (optional)
As you can see, the yellow jumpers are still partially concealed by the matrix, so you may want to extend them to be more accessible. You can use 4 more yellow wires and keep everything linear, use 4 green wires and skip a pin, or use 4 green wires at an angle. I chose the latter, because I didn't have enough yellow connectors and I don't like gaps.
Step 6: Use!
I used this with an arduino uno, and it works great. The wires look like a mess, but that's because I used a bunch of short ones. It might take a little guessing and checking to figure out the right orientation, but I got it working without too much trouble or time.