How to persuade developers to start using feature flag toggles?
Feature toggles are a common practice in high-velocity development because they de-couple development from release.  Dev teams can "soft-release" a new feature to production, in a disabled state.  This allows the feature to be released any time.  If the feature is dependent on other work or preparation, it doesn't have to wait for a major release to go to production.

As far as "convincing" developers to use them, that's an exercise in making the case for the freedom it offers. My experience is that it's not a tough sell to developers.  It's management that tends to be reluctant to try new things.  Try this:


Find a feature toggle framework.  Management/the business may be more
amenable to trying something supported by a common system
Start small.  Introduce the toggle system on a trial basis for a feature that will demonstrate its utility.
Demonstrate the toggle's ability to do A/B testing.  Enable the toggle for a subset of your web farm, then gather metrics on behavior.  Tiny differences in page layout have been demonstrated to have huge effects on revenue for retail applications (e.g. Ebay, Amazon)