I finally returned to Warcraft III several years after purchasing it (and not finishing it). My problem at the time was that I played too many random maps, which ended up being the same thing over and over (settlement building, army building, attack, rinse, repeat) and I got tired of it. I never even finished the Alliance campaign.
In the meantime I played World of Warcraft (which is set four years after the events of Warcraft III) and then I decided to give Warcraft III another try, both to finish a game I never finished and to play through the storylines that led up to WoW. By this time I was familiar with the big characters and events, thanks to playing WoW and reading about the lore, but it was very exciting to actually play through them on my own. Part of the WoW setting is the aftermath of the invasion of the Burning Legion, so to see it all happen in Warcraft III was really quite amazing. While Warcraft I and II seem to be more straightforward (essentially dealing with the humans and the orcs battling each other), Warcraft III really expands the lore of the game world. Arthas falls, the Legion invades, the orcs travel to Kalimdor, the night elves reawaken the druids, etc. There are a lot of big events that you can play through in these four campaigns.
My only complaint would be that sometimes it gets a little repetitive with the settlement building aspect. I know this is the heart of a strategy game, but completing a chapter could take as long as two hours simply because you have to build everything up, create an army, attack an enemy outpost (thereby losing many units), and then re-building the army to do it all again. However, there are many other chapters that do not involve a high degree of settlement building, and some chapters are straightforward exploration and questing. So thankfully there is a variety of play styles across the campaigns.
Finally, regarding the Battle Chest product specifically: this is simply a fantastic value. It comes with separate discs for Reign of Chaos and The Frozen Throne, the original game manuals for each, as well as mini Brady game guides for each. Blizzard certainly did not skimp on the included materials for this collection.