“Links” are the links in your blog’s posts and comments and such which link to your own site or to the images or to whatever else. After the initial import is done, the importer will go back through all the posts and such and correct any of the old links to point to the new ones on the new site. This takes time, and the counter may not update evenly, because not every post or comment may have links in it for it to fix.
The importer doesn’t run continually, but it will run in the background until it finishes. The idea is that you can continue to use your site normally, you don’t need to keep refreshing that page in particular. Any access to any page in WordPress will cause it to kick off the next batch of processing for the importer. Eventually, it will be done. You don’t have to wait for it to finish before you can do things, just use the new site normally.
Otto has described what the links above, these are the internal links between posts. There are 2 reasons to change these. One is that they may refer to a different domain e.g. myblog.blogspot.com but also because the permalinks for the posts is different between blogger and wordpress. It does not change links in the comments.
It is correct to say that the importer does not run continually but you WILL need to stay on the import page to have the import process as it is the javascript on that page that is periodically polling the backend to make the import process.
Otto: We don’t have a Cron for this importer.
My mistake, I thought it was like the Tumblr one.