This is an optional section. If the data for your form field type is just passed along in the POST request like the "native" HTML field types (textboxes, textareas etc), you don't need to specify anything here. The code will simply look at the POST request, pull the appropriate value out and insert it into the database. This is very quick. However, it doesn't handle all cases. The "Saving" tab lets you add custom PHP code that executes on fields of this type, prior to adding to the database, so you can do whatever you want to doctor the field ata.
Form Tools stores all data for a form field in the fields's [prefix]form_[form ID] database column. In some cases, this may require serializing the data, then de-serializing it when viewing / editing.
As a simple example, the Phone Number field type lets users create a phone number field of an arbitrary format. They enter a string like "(xxx) xxx-xxxx" and the field type (on the Displaying -> Edit Field tab) converts that into three textboxes. So, thinking as a developer, that means that when a form containing these phone numbers is submitted, the POST request will include an arbitrary number of textboxes (depending on the string they entered) - each containing a part of the phone number field. So in order to store the entire phone number in the (single) database field, you need to piece it all together and then submit the concatenated - or serialized - result. And that's precisely what's being done in this field type, shown in the screenshot above.
The phone number stores all the phone number parts, separated by the pipe character, so it's easy to separate and re-create when viewing / editing.
The $vars variable contains all the information about the POST request and field - including whatever arbitrary settings were specified for the field type. To see what's available, the simplest thing to do is just do a print_r($vars); exit; in that section, then in the Form Tools interface, submit the Edit Submission page on a View that contains this field type. That will display on the screen the entire contents of the variable.
You MUST define a $value var
One last tip: if you enter custom PHP code here, you must define a $value variable that contains the value to store in the database. It should be a string or number, nothing else. Arrays will just be converted to "Array()" which won't help anybody!