Firediff is a Firebug add-on that tracks changes done in Firebug. It logs everything you'll do in the HTML pane (great) but also your brief use of the Web Developer Toolbar extension (not so great), say Shift-Ctrl-F to obtain a font-size information in px.
I have seen a Firebug extension in Chrome but didn't test it, I use Firediff with Firefox.
Excellent product. Sounds like exactly what you're looking for and much more.
EDIT: Several other answers here have mentioned Google Chrome's ability to link to your local files (which is very very cool). Check out the other answers!
I have found the answer to this, at least as of Chrome v14.
While in the Elements section, just click on the "filename:linenumber" link next to the CSS rules. The CSS file that shows up will contain all of the modifications.
If you edit external CSS, then you can drag its latest revision out of the Resources panel into any text editor that supports DnD (see http://www.webkit.org/blog/1463/web-inspector-styles-enhanced/, the "Persisting Changes" section for more detail.) You can also revert your CSS changes to any earlier version of the stylesheet resource (in the right-click popup menu of any stylesheet revision.)
As mentioned by cloudworks, the answer to this has changed. This can now be accomplished rather well by the Chrome DevTools Autosave extension. This tool tracks CSS and JavaScript changes made within the Chrome Developer Tools console, and saves them back to local files. For instructions to install and setup the extension, please refer to the guide written by @addyosmani on his blog, here.
There is also a handy screencast which details the extension rather well.
If you're using the Firefox stock dev tools you can edit the css directly in the tools dialog - click the CSS viewport button (that's the button at the top with the {} symbol) and edit your css directly. It will update in realtime in the browser and when you're done just copy-paste it directly into your css file. Nice!
To add an answer for Safari specifically — it's kind of possible.
When you edit CSS in the Styles section in the Inspector for an existing CSS file, you can hit Cmd-S to re-save the entire file with the changes. However, if you're using a meta language like Sass / preprocessor / generating your CSS with bundling etc, I don't think this really solves that problem, though it may be possible with CSS source maps.
When you edit CSS at the top of the Styles section, under Style Attribute to add inline styles (not tied to an existing CSS file), it doesn't seem possible to easily export all of those changes. For now, I'm just copying and pasting the overrides manually for each element.
In Chrome, in the css inspector you can click and hold the + button, then choose to add your changes to the inspector-stylesheet. It's not as convenient as directly editing in your css-selectors, but what you write will all be in inspector-stylesheet.css
In Chrome there is also the Changes tab in the console drawer that displays all the modifications of CSS. It's not an export, but at least it is very convenient to quickly grasp what has changed.
Both Firefox and Chrome support this feature now, but worth to note that in some platforms if not all Chrome does not show it by default, you need to enable the "Changes" view to see it (in my Kubuntu Linux 20.04 it wasn't by default), here is how you can enable it: go to the "Customize and Control DevTools" button in the Developer Tools bar > "More tools" > "Changes", then the tab will appear at the button:
In Firefox there is no need to enable it, but if you come from the Chrom* world may be hard to find it. Just check the last section in the right at the "Inspector" tab: