The simple answer is that the United Nations uses a gaggle of translators, but the longer answer is simpler than that. The six official languages of the United Nations are Arabic, Chinese, English, French, Russian, and Spanish. If a delegate addresses the United Nations in one of these six languages, they will be translated into the other five languages.
Official U.N. documents are also normally issued in all six official languages. The United Nations Website is also maintained in all six languages.
A delegate or guest who wants to speak to the United Nations in an unofficial language must provide either an interpretation or a written text of the speech in one of the official languages.
The original U.N. Charter of 1945 was issued in Chinese, English, French, Russian, and Spanish, which became the official languages of the U.N. the following year, with English and French as the "working languages." Arabic was added as an official language in 1973, and the "working language" designation was eliminated soon after.