It's a (Wo/Hu/Non-Hu/Meta-Hu/Über/Super)Man's Life in the Modern Army! Short-lived Imperial Army recruitment slogan
The Military sphere is designed for Houses specializing in the art of war. With this sphere, Houses are given perks to their military prowess and recruitment power, allowing them to make income from educating recruits at Military Academies and from hiring out legions as mercenaries.
There are two principal reasons for doing this. Firstly, you will be able to construct Military Academies in order to produce higher quality troops on a planet and earn income from recruitment and military education there. If you have a Military Academy on a planet and you recruit a unit to be based in any of your fortresses on that planet, the unit will be of 'Standard' quality rather than 'Rookie' quality as would normally be the case. As concerns income, each time any House, including your own, recruits units on a planet with one or more Military Academies, a percentage of the recruitment cost is earnt as income for the Academies. The exact amount depends on the number of Academies present, but will never exceed 20% of the recruitment cost. Additionally, all Academies will earn a small amount of income from educating officers and private individuals on the planet concerned. The amount earned depends upon the size and population of the planet, and also on the number of dynasty members of Houses for whom the planet is homeworld that are sent to Military or Elite Academy.
The second advantage to operating within this sphere as a speciality, is the ability to offer out your units for hire as mercenaries, thereby generating income over and above that gained from Academies. Should the unit survive it's mercenary service with the hiring House, you will have probably covered the cost of recruiting the unit in the first place, and the unit may also have gained some experience.
For full details of hiring out mercenaries, see the section within the rules for Advanced Warfare.
As seen in the Rulebook
At first glance, there's little reason to build a military academy on more than just your HQ, as recruiting units will always cost less on your home planet - even after the cost of transferring the unit to the fortress you want them to man. Military academies will rarely make enough money to cover even their own maintenance costs, especially if there is even one other house in your range with the military sphere. However, having military academies on multiple planets allows you to continue to build Standard troops even if the academies on your home planet are destroyed, and the income you gain from military academies on other Houses' planets give you valuable intel on when they are building forces.
Additionally when offering mercenaries the amount you charge is reduced for every defence concession you have as your base recruitment cost is lower.
On Mercenaries: Be aware that there's nothing stopping a house that hires your legions from using them against you. It might be prudent to limit who can hire legions from you to allies.
To clarify the above: while you cannot attack someone with mercenaries you have hired from them, there's nothing stopping you from using them in a defensive caliber against anyone. In short, it is unwise to hire out mercenaries to someone you intend to attack later.