Traditionally, libraries would buy copies of physical books and lend them out. As these books are physical, only one at a time could be lent. Libraries were not allowed to make photocopies of books and lend out multiple at a time.

When ebooks came along, making digital "photocopies" became potentially much easier. Thus, many ebooks came with DRM attached to prevent copying. As digital rights are different from the rights to physical goods, authors and publishers would generally provide a license for lending of ebooks in exchange for a fee. Libraries could still buy ebooks and lend them out, but the number of times they could lend them was restricted based on how much they paid for those rights.

The Internet Archive came along, bought a bunch of books, made ebook versions of them, and then lent them out -- usually one at a time, but for a while they lent out unlimited copies. Their argument is that buying a physical book once should allow them to lend it in ebook form one at a time, just like it allows them to lend it in physical book form one at a time, without paying any licensing fee for those electronic rights.

The Internet Archive should survive because it does a lot of good and useful stuff. It will survive even if it loses this case. At issue is whether or not this particular lending library practice should survive. Those who argue that it should generally don't think ebooks should have any copying restrictions anyway and think everyone should be able to get any book for free without paying the authors or publishers anything, because they see publishers as already too rich and too powerful and evil, and they believe authors will benefit more from the "increased exposure" of freely pirated ebooks and more people will buy their books as a result. They are generally also the same people who think copyrights are too long anyway and think that long copyrights only serve to benefit the publishers and not the authors.

People who support individual rights, authors, and publishers are generally against it, because they believe digital lending rights are different from physical lending rights and this is an important revenue stream for both authors and the publishing industry. Creating a new right that allows ebook copying not only denies individuals a right over the control of their work, but hurts them financially. They believe libraries are doing just fine with the current lending scheme and that there's no need to create a new giant free ebook library.