The phrase "to promote the initiative" means to support and encourage the idea or action plan, and to spread awareness or make it known to others. Full definition
We want to create an international reach as well as promoting initiatives which will support event participants towards sustainability. (ethicalfashionforum.com)
The approach promotes initiative, curiosity, persistence and cooperation. (ncsheadstart.org)
This is likely to require extra time and resources, but researchers promoting the initiative argue it is worth the effort. (sciencemag.org)