The last live date tells you the last time the link’s status was OK. Links that were never live in the first place, like prospect links for instance don’t have this date of course. Links that were already in error when this feature was released (18 Dec. 2013) don’t have this information either.
Let’s see a few cases where the last live date is very useful:
Your have a paid link that has been removed by mistake. You have contacted your partner and asked them to correct the problem and put back the link. With the last live date you can know how long the link has been missing and put pressure on your partner. It is particularly important to have this kind of problem corrected as soon as possible, before search engines notice, as a link that is removed and put back again is an indication of a negotiated link.
The source page cannot be crawled because the website is down. Sometimes, websites can go momentarily down during maintenance operations or some technical issue. But websites need to be back up again as soon as possible. A website that stays down for several days is a good indication that it may never be back again. Web businesses go bust sometimes and although the domain may still resolve, the servers may just not be maintained any more.
The link gets a 50x http error. The website is up but that very page gives this error because of some bug on the server. The webmaster may just not have noticed the problem because the rest of the website is OK. Just contact the webmaster and inform him about the issue.
