The Chrome Privacy Whitepaper gives background detail regarding the features which this API can control.

Manifest

You must declare the "privacy" permission in your extension's manifest to use the API. For example:

{
  "name": "My extension",
  ...
  "permissions": [
    "privacy"
  ],
  ...
}

Usage

Reading the current value of a Chrome setting is straightforward. You'll first need to find the property you're interested in, then you'll call get() on that object in order to retrieve its current value and your extension's level of control. For example, to determine if Chrome's Autofill feature is enabled, you'd write:

chrome.privacy.services.autofillEnabled.get({}, function(details) {
  if (details.value)
    console.log('Autofill is on!');
  else
    console.log('Autofill is off!');
});

Changing the value of a setting is a little bit more complex, simply because you first must verify that your extension can control the setting. The user won't see any change to her settings if your extension toggles a setting that is either locked to a specific value by enterprise policies (levelOfControl will be set to "not_controllable"), or if another extension is controlling the value (levelOfControl will be set to "controlled_by_other_extensions"). The set() call will succeed, but the setting will be immediately overridden. As this might be confusing, it is advisable to warn the user when the settings they've chosen aren't practically applied.

Full details about extensions' ability to control ChromeSettings can be found under chrome.types.ChromeSetting.

This means that you ought to use the get() method to determine your level of access, and then only call set() if your extension can grab control over the setting (in fact if your extension can't control the setting it's probably a good idea to visually disable the functionality to reduce user confusion):

chrome.privacy.services.autofillEnabled.get({}, function(details) {
  if (details.levelOfControl === 'controllable_by_this_extension') {
    chrome.privacy.services.autofillEnabled.set({ value: true }, function() {
      if (chrome.runtime.lastError === undefined)
        console.log("Hooray, it worked!");
      else
        console.log("Sadness!", chrome.runtime.lastError);
    }
  }
});

If you're interested in changes to a setting's value, add a listener to its onChange event. Among other uses, this will allow you to warn the user if a more recently installed extension grabs control of a setting, or if enterprise policy overrides your control. To listen for changes to Autofill's status, for example, the following code would suffice:

chrome.privacy.services.autofillEnabled.onChange.addListener(
    function (details) {
      // The new value is stored in `details.value`, the new level of control
      // in `details.levelOfControl`, and `details.incognitoSpecific` will be
      // `true` if the value is specific to Incognito mode.
    });

Examples

For example code, see the Privacy API samples.