When you share a link on Facebook, the link is displayed in your Facebook post along with the page title that Facebook retrieves from your site. However, if you've recently updated the page's title, you may find that Facebook continues to show the old title. This occurs because Facebook is taking the title from a stored (cached) copy of your page.
One way to prevent this from happening is to tell Facebook to reload the page before you share the link. You can do this by entering your page URL into the Facebook Debugger Tool and clicking "fetch new scrape information". Facebook will then reload the page and show you a preview of what your shared link will look like. When you're happy with how it looks you can go ahead and share the link via Facebook.
I'm not aware of a way to tell Facebook to update the titles on previously shared links, however.