The War Academy is unlocked for your kingdom at Castle level 3, and is the building where you can upgrade the overall stats for all of your heroes by using your Art of War books. 
Bottom Line Up Front[]
- Investing in your War Books early is always worth it. In association with the Precious Gems event, if you can get 25k clan coins, it's almost a no-brainer that you can spend 23,200 coins + 8.7 mil gold to level up all of your hero classes to half-full.
- The best thing you can do for yourself BEFORE choosing to refresh: max out all of your War Books purchases AT A MINIMUM.
- Of course also buy whatever else you want from the Clan Shop; it's probably the best shop in the game that has some of the most powerful heroes and equipment available.
Upgrading your Hero Classes[]
This is what your War Academy looks like.
The Academy covers all 5 of the Hero Classes in the game. 
The rank is in the middle; and based off of previous game versions, you can rank up based on how many times you max out your entire academy (wood, bronze, iron, silver, gold, etc).
The Arts of War[]
For each hero class, there are 8 arts that you can upgrade with war books. Leveling these arts permanently increase stat bonuses for every hero of that class in terms of attack, defense, and health.
NOTE: These bonuses are the same for all 5 hero classes! Same thing applies for cost, which finally leads us to the tables (for rank 1):
Cost Tables[]
the Puzzle Breakers War Academy spreadsheet
Now remember, each art can be unlocked when the previous art is upgraded to level 5. So in the above table on the right, the calculations for "half-full" basically show your bonuses, if all 8 arts are unlocked (but not fully upgraded) for a single hero class.
this is the class that is half-full with all 8 arts unlocked, and the summary of stat bonuses on the right. Please note, the yellow "5.5%" in the summary is the true stat bonus for a half-full arts upgrade (I accidentally bought another upgrade before the screenshot).
Buying War Books[]
Art of War books can be bought from the Clan Shop:
For more information, please visit the Clan Shop page.
Team Power Upgrade[]
So, how much does this affect your heroes stats/overall team power?
Quite a lot! Credit to (ง'-')ง Antony for tracking his example, where his team power went up ~900 points just from upgrading all of his hero types to half-full in the Academy.