Bonjour,

Voici les points les plus importants à corriger (rouge) et améliorer (rose) pour la première moitié du texte.

Some encounters can change somebody’s [your] life. Meet[Le fait de rencontrer => v-ing] people permit [choisir un autre verbe] to know s culture [si vous êtes de culture différente!], enrich (s’enrichir d’un point de vue de la culture) and so to see the things in a different way. This essay will describe [utiliser le présent, à moins que ce paragraphe n'annonce une dizaine de pages] a personal experience what is the encounters with people that I thought have nothing in common with them.

Eight months ago, I see[prétérit obligatoire avec ago] arrive ['voir arriver' = langage trop familier pour un 'essay' => venir vers moi] two persons who were quite old [older then you or elderly?] and I don’t expect [=> c'est un passé antérieur!]. They are [passé] called Alfred and Jean-françois [Vous passez un peu vite de l'inconnu au connu, non ?] . I play [Comment le savaient-il ? Si vous étiez en train d'en jouer, alors dites-le] accordion and [they asked me if I would [verb missing: would like to join them to] ] form a part of an orchestra. After have [-ing] hesitated for a small moment, I say [passé!] « Yes ». And there [this is how]  everything begann…

expressions that might help:
be culturally enriched
be enriched by other cultures
be enriched by others' experiences and ways of thinking
be enriched by cultural differences
be enriched by learning from the culture