The first phase is the drawing of brine from the Salt tank through the resin.
In this particular case further downward movement of the control valve causes the piston to realign different ports to water flow. At the same time a second valve, connected to the brine tank opens allowing the brine to mix with the incoming water and flow onto the resin.
You may recall this method of water softening was termed "Ion exchange" and that the the Calcium & Magnesium in the water was attracted to the resin, exchanging with Sodium. In this case the same "Ion exchange" process is used except this time in a reverse sense. Now the high Sodium in the Brine solution is attracted to the resin exchanging this time with the collected Calcium & Magnesium, which is discharged to waste.
The reason this part of the Regeneration is so long is that the brine made to pass through the resin at a very slow rate. The longer period the brine is in contact with the resin the more effecient the Regeneration.
Ideally this part of the sequence would take 25 minutes or less, again depending on the age of the Softener.
The second phase of the sequence is a slow rinse, and is described next.

The second part of the Brine & Rinse operation is as you might expect, the Rinsing part. You will notice from the second diagram the brine in the Salt tank has been drawn throught the resin and a small ball has closed the system down. This is to prevent air being drawn into the softener. You will notice also that now only fresh water is being used, this fresh water rinses away the salt, but only slowly so that salt/resin contact time is as long as possible. Ideally this second part takes the same length of time as the Brine draw part.