Hi and welcome to mobileread
Books that you check out from online libraries are all protected with something called DRM - Digital Rights Management. Some ereading devices work out-of-the-box with libraries such as Sony readers and I'm afraid that the Kindle does not.
You have a number of choices really - exchange your Kindle for a device that works with online library lending, find out how to remove the DRM from the ebooks which as Harry pointed out may be illegal OR accept that you will have to just read library books on your PC or you can also now read them on Android phones and iPhones.
You can get plenty of free books for your Kindle - Amazon provide free classics and also some modern free books (quite a few every week if you live in the US) and there are also free classics here on mobileread and also at places like manybooks.net and feedbooks.com
Reading doesn't have to cost the earth