The answer for this question can be broken down into two parts:
Meaning and process of Ashvamedha ritual.
Why was it performed?
Solution:
The raja performong the Ashvamedha let loose a horse to roam around in the surrounding areas; during the ritual.
If the horse was able to pass through any raja’s territory; this meant that that particular raja accepted the supremacy of the ambitious raja.
If the horse was not allowed to pass, the Raja of that kingdom had to fight with the ambitious raja’s army.
The winner of that battle was to become the ultimate raja. After the horse used to complete its round of all the territories; other rajas were sent invitation to attend the yajna. This used to mark the beginning of supremacy of the raja who performed the ashwadedha yajna.