The phrase "to support this blog" means to provide assistance or help to keep the blog running by contributing financially or by engaging with the content in a positive way, such as sharing it with others or leaving comments. Full definition
Thank you so much for supporting my blog by using these links to purchase items at no extra cost to you. (themamaknows.com)
These links don't cost you any more if you purchase through them but I will get a percentage of the sale, thank you for helping support our blog in this way. (findingsilverpennies.com)
You are of course free to sign up for these offers outside of these links provided but you help support the blog by using them. (travelpointsguide.com)