West Semitic is the division of the Semitic family holding Ugaritic, Phoenician, Hebrew, Aramaic, Arabic, the Ethiopian languages including Geez, and the Old South Arabian group, as against East Semitic, which is Akkadian and its dialects. Robert Hetzron's edited survey describes the branch and the arguments about its internal shape, which are not settled.
The reason it matters to the atlas is arithmetic. The Hebrew Bible is in Hebrew and Aramaic, the Targums and much of the Talmud are Aramaic, the language Jesus spoke was Aramaic, the Peshitta and the Syriac fathers are in a form of Aramaic, the Quran is in Arabic, and the Ethiopian Orthodox canon is in Geez. Edward Ullendorff's comparative work on the Ethiopian languages covers the southern end of that range. One branch of one family carries the primary scriptures of Judaism, of Christianity in several of its oldest forms, of Islam, of the Samaritans, of the Mandaeans and of the Druze.
The languages are close enough that the traditions can read one another. Medieval Jewish grammarians developed the analysis of Hebrew using tools built for Arabic, and Jewish philosophers wrote in Arabic about Hebrew scripture, which is a direct consequence of the family relation.