having a deep knowledge of one's own country's culture
being well-informed about the culture of your country
yuhkiさんの文章を全部英訳にすると、こうなります:
Being well-informed about your own country's culture makes you secure in your identity as a citizen of that country, and as an international person, you can then have deep interactions with people from other countries.
「自国の文化に精通することで、自国人としてのアイデンティティが確立され、国際人として他国の人と深く交流できる」というのは Being familiar with your own culture establishes your identity as a native, and as a citizen of the world you can exchange things on a deep and profound level with people from other countries と表現できます。
例文 Being familiar with other cultures let's you understand your own culture well.
「ほかの文化を精通するのは、自分の文化をよく知られる。」
参考になれば幸いです。