How does 海报 became "a poster"?
Baidu says:

海报这一名称，最早起源于上海。旧时，海报是用于戏剧、电影等演出或球赛等活动的招帖。上海的人通常把职业性的戏剧演出称为“海”，而把从事职业性戏剧的表演称为“下海”。作为剧目演出信息的具有宣传性的招徕顾客性的张贴物，也许是因为这个，人们便把它叫做“海报”。

Which means, if I summarise:
The word originated in Shanghai, where people referred to professional theatrical productions as "海". Hence "海报"