why in the military was the APO/FPO mail system used?

Until 1980, each U.S. military service managed its own military mail program. In 1980, the Department of Defense (DoD) designated the Secretary of the Army as the single military mail manager. The Military Postal Service Agency (MPSA) was created to perform this task, with a jointly-staffed headquarters located in the National Capital Region. MPSA is required to adhere to United States Postal Service (USPS) rules, federal laws, and various international laws and agreements for movement of military mail into over 85 countries. MPSA is the single DoD point of contact with the USPS. It conducts DoD contingency planning and provides postal support to theater Combatant Commanders through Services' theater postal commands. It serves as a proponent of DoD's cost control policy for DoD official mail and as the functional director for military mail. MPSA also monitors transportation funds dedicated to overseas mail movement.

In addition to supporting U.S. military installations overseas, the MPSA also supports mail delivery to and from U.S. diplomatic facilities abroad and, of course, to deployed personnel in combat zones worldwide. In most (but not all) locations, special services such as Express Mail, registered mail, certified mail, money orders, etc. are offered. Mail sent from one military post office (MPO) to another MPO in the same theater (e.g., mail from an MPO in Japan to another MPO in Singapore) can be sent for free on a space available basis. Because the DoD provides or pays for overseas transport for U.S. military mail between the U.S. and overseas duty stations, as well as delivery services in theater, military postage rates are equivalent to U.S. domestic postage rates. All military mail between the U.S. and overseas locations is subject to customs inspection in the country of destination, and customs declarations must normally be attached to packages and larger mail pieces. Some host nations may restrict or prohibit the mailing or receipt of certain items, such as pornography, meat products (especially pork), firearms, tobacco, etc., via the U.S. military postal system. Other host nations (e.g., Germany) may have a special interchange agreement with the military postal service allowing for the mailing of articles to host nation addresses at U.S. domestic postal rates.

In accordance with treaties and other international agreements, use of U.S. military postal services overseas is normally restricted to authorized customers, i.e., U.S. Government personnel (military, civilian and diplomatic) stationed overseas (either permanently or on temporary duty orders) and their dependents. Further restrictions may exist on a country-by-country basis depending on the host nation involved and the status of the postal customer. Retired U.S. military personnel living overseas may also have limited access to the military postal service, depending on their country of residence. The only time U.S. citizens not associated with the U.S. Forces may use military post offices is for the sending of absentee ballots.[1] The easiest way for U.S. Citizens not affiliated with the military to obtain access to the military postal service while overseas is at a U.S. Embassy, as access to other overseas facilities may be restricted.

Overseas military post offices operated or supported by the Army or Air Force use the city abbreviation APO [Army Post Office / Air (Force) Post Office], while overseas military post offices operated or supported by the Navy, Marine Corps or U.S. Coast Guard use the city abbreviation FPO (Fleet Post Office). Three quasi-State codes have been assigned depending on the (approximate) geographic location of the military mail recipient and also the carrier route to be taken. They are:

AE (Armed Forces Europe / Canada / Middle East / Africa) - 09XXX

AP (Armed Forces Pacific) - 96XXX

AA (Armed Forces Americas) - 34XXX

FPO is Fleet Post Office for Navy & Marine Corps personnel or it can also be used by Army, Air Force and Coast Guard personnel stationed at a Navy or Marine Corps Base. APO is for Army Post Office and is mainly used by the Army & Air Force when overseas. It can also be used by the Navy, Marines and Coast Guard when stationed on a Army or AF base/post.

ETA: Actually USPS does still operate some of the overseas base/post Post Offices, the civilians hired and staffed there are USPS Government workers.

I don't think you send mail to the base. I think you send it to an APO in New York and they forward it to the soldier, so I think it's just domestic postage rates, but don't quote me on that.

They are still used

APO = Army Post Office

FPO = Fleet Post Office

The US Mail does not deliver overseas

Still used too.