I have looking at these type of projects and would like to build one.
There are quite some complicated design available but I want a simple low cost one.
I have read a bit about multiplexing. But I am not sure about how everything works with each other.
Is it possible to build one using the Arduino Uno and 74HC595 (Shift Registers)?
Yes, but it will perform poorly. The HC595 was not designed to sink & source a lot of current.
You will have better results using cd74AC164 to source current to columns, and N-channel, low Rds, Logic Level MOSFET to sink current from each layer. The HC595 can be used to control the N-channel MOSFET.
Send the data for a layer to 8 '164s, turn on the MOSFET for that layer.
Wait 4mS (1/30/8), turn the MOSFET off and repeat for the next layer.