Brid.gy is great, but if I'm understanding your question, I'm not sure how it could be used to display your Webmentions on your site. Anything is possible, but that functionality may be way out of its scope. It's primarily used to do two things:
1. Send you webmention notifications from various silos like Twitter. Once you receive them, then you'd still need a way to save and display them if you wish.
2. Accept incoming notifications from your original posts as a means to cross-post your content to social media platforms