Spiritual science in this context is also known as 'anthroposophy'. Anthroposophy was described by Rudolf Steiner, founder of the General Anthroposophical Society in 1923, in the following way:
By anthroposophy I mean a scientific investigation of the spiritual world which looks beyond the one-sided approaches of mere science or the usual mysticism. Before anthroposophy attempts to penetrate the supersensible world it develops in the soul of the researcher the faculties not yet active in everyday consciousness or conventional science which enable such a penetration.
(Rudolf Steiner, from 'Philosophy & Anthroposophy', collected essays 1904 - 1923, Rudolf Steiner Verlag, 1984, GA35)