So, why should you check them? You'll want to make sure that these inbound links are pointing to existing pages on your website. Some of these links might be old, so the pages they are linking to may have been moved or even deleted from your website. The importance of that link is wasted if the search engines' spiders cannot get to that page and review the content. Maybe you were considering removing the page but find that it is being heavily linked to - you may want to reconsider the move and leave the page if other websites and users find it to be useful content.
Now that you've checked your backlinks and found links that are pointing to pages that have moved or been removed, what do you do? You have a few options. First, and best, contact the website to let them know the page moved and ask if he or she would mind changing the link to the page's new location. If the page was removed, you could suggest another page that would be relevant to their users and content of their page. What if you are unable to reach the website owner and there are a number of links pointing to the old page? Your second option is to set up a 301 redirect to point the incoming link to the new page's location so users following the link can still get to your content instead of receiving a "page not found" error.
There are many free and paid tools you can use to check your backlinks; one of our favorite tools is Open Site Explorer. Just do a search for "backlink checker" and choose the tool that works best for you!