Reference: Balakanda 8th Sarga Why did Dasharatha propose performance of Ashwamedha to beget children? Ashwamedha is primarily to establish supremacy over other kings. I am unable to understand the logic or dharma behind this.
I am quoting my friend Sri Kiron Krishnan's answer here, on Aswamedha in other site, to explain the meaning of the same as per Vedic mantras. According to Vedic sense, Aswamedha should be viewed in an esoteric manner only, but not physical sacrifice of Aswa or Horse.
By the time Ramayana was composed, please do not get confused with the period of occurrence of Ramayana, this got degenerated to physical sacrifice. However, it was being done in the times of King Dasaratha to beget sons only, but not for establishing supremacy over others.
King Dasaratha proposes to perform Horse sacrifice for begetting sons, as he was not having sons.
मम लालप्यमानस्य सुतार्थं नास्ति वै सुखम् | तदर्थं हयमेधेन यक्ष्यामीति मतिर्मम || १-८-८
"My mind is tumultuous without quietude for I have no sons... for that reason, I wish perform Aswamedha, Vedic Horse Ritual... this is my thinking.
By the time Mahabharata was composed, the purpose of Aswamedha had become to purify one.
Yudhishthira replied, 'Beyond a doubt, the Horse-sacrifice purifieth princes.
The Horse in Mahabharata period was released for conquering other kings, and to display the glory of Yudhisthira.
"Vyasa said, 'Myself, O son of Kunti, and Paila and Yajnavalkya, shall without doubt, achieve every rite at the proper time. The rite of initiating thee will be performed on the day of full moon belonging to the month of Chaitra. Let all the necessaries of the sacrifice, O foremost of men, be got ready.
Let Sutas well-versed in the science of horses, and let Brahmanas also possessed of the same lore, select, after examination, a worthy horse in order that thy sacrifice maybe completed. Loosening the animal according to the injunctions of the scriptures, let him wander over the whole Earth with her belt of seas, displaying thy blazing glory, O king!'