SamAshrayanam means ‘to approach (AchAryA) with all sincerity’. In a nut shell, during ‘SamAshrayanam’, the AchAryA initiates a person, irrespective of caste, creed or sex, as his sishyA. It is a commitment from the disciple that he or she will live as per the wishes of the AchAryA. Thus the person gets the link to the Sri Vaishnava paramparA. During SamAshrayanam, Pancha SamskAram (five Purifications) is performed and he becomes a “Sri VaishnavA”.
Here is an article based on a dialogue about Pancha Samskara/Samasrayanam, written by Sri U.Ve. Purusai Nadathoor Krishnamacharya Swami, and translated in English by Sri U.Ve. Kurchi Narayanan Swami.