Let's break the question down into parts.
Autosave uses a modified version of the jquery.field plugin (modified since it doesn't support Drupal field naming conventions very well)
This means that when the project was written, they took a plugin (piece of code) from another module and used it as a base for their own code.
This version of the plugin was based on jquery 1.2 and does not work with jquery 1.3.
This bit taken just by itself means that the module will not work if you are using JQuery 1.3 or higher with your Drupal site. It does, however look like old information and should be disregarded because:
Rev 6.x-2.7 DOES support jquery 1.3 and should still support jq 1.2. Please raise an issue if you find that it doesn't.
...which states that version 6.x-2.7 supports JQuery 1.3.
The module has been updated several times since then, as you can see the Drupal 6 module is now on version 6.x-2.11. It is likely that the Drupal 7 version of the module supports much more recent versions of JQuery since it was last updated 6 months after the last Drupal 6 version.
What should you do?
- Make a development copy of your site if you haven't already, where you can break things without worry. Use a VirtualBox or similar if you know how or can learn.
- Install the current version of the module and see if it works. Click around your site and see if you can find any unexpected behaviour
- If it doesn't work, consider installing JQuery Update which allows you to choose the version of JQuery being used by your site.
