The Shrine of Odin is an essential building of your town, focused on your military stats. You can only have one in your city. It is mandatory to upgrade the Shrine of Odin if you want to upgrade your Palace.
The shrine of Odin benefits are as follows :
Allows you to check what are your troops available in town, your reinforcements, your actives marches and garnisons;
Upgrading the Shrine of Odin lets you increase the number of troops you can send on a march which is essential for attacking or generating resources; The maximum amount of troops can be extended by boosters like "Marches" (several different types) and by your VIP bonus, up to 10% once you reach VIP level 10, so level 21 gives you 125k (according to table below) which gives 137.5k troops in a single march with VIP Bonus level 10 or more before Marches boosters.
upgrading the Shrine of Odin also allows you to increase the number of marches you can have simultaneously from 1 to 5 at level 21. Note that two skills from the Oracle also allows you to increase the maximum number or simultaneous marches.
Requirements : you need a Palace of the same level in order to upgrade the Shrine Of Odin.