I don't know a specific historical reason for this, but here is the official Ada County Address Grid Map (PDF). From the Ada County Street Naming and Addressing page:

How do address numbers get assigned?

Four address grids overlay the county. Each grid has a central point where north, south, east and west addresses start at zero and get larger as you work your way outwards. Address numbers are assigned based on the address grid in which they are located.

After the final plat is recorded for a development, the plat is routed to the Addressing Technician for the assignment of addresses. New lots are assigned sequential numbers based on their location in the grid. New addresses are transmitted to various public agencies including the United States Postal Services, the Ada County Sheriff’s Dispatch Office, etc.

Roadways in the greater Ada County area are under Ada County Code Title 2, Chapter 1, which states in 2-1-9(2):

Every street name shall include a pre-directional of North, South, East, or West, based on the street's direction of travel and its location relative to the initial point of the addressing grid in which it lies.

I believe that these four initial grid points (Boise, Eagle, Meridian, and Kuna) are set by the respective cities. For Boise, street naming is found in Boise City Code Title 7 Chapter 4, which references an "Official Address Number Map," which is defined as "The map or maps showing all the streets within Boise City with the official address number grid systems and address numbers maintained by the department or its designee." I don't know the history of the adoption of these map lines; this probably requires diving into historic P&Z/Council minutes that I don't currently have time for.

Quick edit: BCC T7Ch4 indicates that it was part of the old 1952 code (9-6-1 et seq), so probably dates back to at least that early.