The phrase "to follow this blog" means that you want to subscribe to or keep up with updates from a specific blog. It allows you to receive notifications or updates whenever new content is posted on that blog. Full definition
I just recently starting following your blog for health reasons but wanted to pass along a few adoption agencies that are wonderful... my four kids are adopted. (glutenfreegirl.com)
I've been following your blog posts for years. (justinhealth.com)
I'm following your blog with great interest because I am doing a kitchen right now. (mariakillam.com)