Using the sponge is easy enough. Just remove it and get it wet. Make sure to squeezes as much water out as you can so that is is only damp. Then replace it back in the holder.
When soldering periodically wipe your soldering tip on the damp sponge to clean it.
Using a brass sponge requires no water. Just push your soldering tip up and down into the brass sponge.
You may need to periodically remove the brass sponge from the bass and empty out the left over solder bits.
Now, which one is the better choice? The damp sponge can put stress and wear on the soldering iron tip by cooling it down. You may also need to wait a little longer before you go back to soldering as the tip temperature will have cooled a little.
With the brass sponge there is no need to worry about the soldering iron tip becoming cooler as the brass sponge is not wet. Usually we use the brass sponge.
Both the regular sponge and the brass sponge should be replaced periodically when they become worn out or dirty.