speak one’s language

The Washington Postにあった記事からの引用です。
以前にブログで採り上げられたspeak the same language（同じ考え〔趣味〕を持っている）という表現からもわかるように，languageには「言葉」だけでなく，「考え」を意味します。
cf. http://d.hatena.ne.jp/A30/20100521/1274416511
英英辞書でもlanguageをsigns, movements, or sounds that express ideas or feelings（LDOCE）のように解説していました。よって，小見出しにあるspeak their languageは「自分たちの考え（あるいは存在感）を示す」と解釈できます。（Koyamamoto）

A familiar place to age
Foreign-born seniors in the United States are finding more centers that speak their language on culture, meals and leisure activities
If Mohammad Rahro, 91, had gone to just any senior center in Maryland, chances are he would not have encountered someone who remembered the headmaster with four wives who rode to school each day on a beloved white donkey. Instead, on his first day at the Loving Care Adult Medical Day Care Center in Gaithersburg, he ran into Ali Ameri, 96. Rahro’s hair was sparse and he leaned on a walker, but Ameri recognized him immediately — the two were schoolmates in Tehran in the early 1930s.
http://thewashingtonpost.newspaperdirect.com/epaper/viewer.aspx