matgb mentioned reasons for the tracking system as a whole, but (as I was there at the time) I can tell you precisely why LJ introduced the inbox in particular: many email providers would eat mail from LJ, or treat it like spam, because LJ sent out so much email that many providers thought the high volume indicated they/we were spammy spammers who spammed. So, the inbox was a method for collecting all your comment replies/notices of things you wanted to be notified about in addition to having an email sent to you, in case the email went astray.

Many people, myself included, use the inbox as a "oh yeah I wanted to reply to that" collector -- I clean out my inbox once every few days and delete the comments I know I'm not going to reply to or the comments I know I've already replied to, and what's left is essentially a to-do list.