The phrase "to promote this giveaway" means to spread the word or advertise about a specific event or activity where prizes or gifts are being offered. Full definition
You can promote the giveaway on your blog or newsletter, and the other authors will do the same, sending highly targeted visitors to your site. (webdesignrelief.com)
Giving stuff away is not free, all the more if you're promoting the giveaway through advertising. (blog.reedsy.com)
I always promote my giveaways as long as they run. (glamdevils.com)