It could be that your link checking tool just updated it's index - or that you're using two different tools which have different sizes of index. Also make sure that the settings are the same on both reports.
Check that your tool is set to external sites only. If it's counting internal links then *facepalm* and carry on.
It could eaisly be that one (or more) of the sites that link to you has duplicated the link on dynamic pages. Export your data to a spread sheet and get a count of the number of domains linking to you.
Try to figure out where the links are coming from. Is a scraper distributing your content across a network of spammy sites? Or are they all appearing as footer/blogroll links in low quality blogs?
I have no idea where all this traffic has come from...
Have you experienced more traffic?
Re the links, I agree with BenFox. But visually inspect the list of links to see if there's anything strange just in case. It could as BenFox said, perhaps a handful of blogs with sitewide links just had more of their outbound links counted.
This happened to us last year. It was scraper sites copying our blogs and articles to use as their own unique content. It didnt hurt our rankings but it didnt improve them either. Our rankings stayed where we expected them which was in the top 20 for most of our primary keywords. I would not worry about this. As long as you arent linking back to them you should be ok.