Tadarthah for that purpose, for that sake
Drsyasya of the seen, of the knowable, nature (prakrti)
Atma seer (purusa), soul, principle of life, awareness, 'witnesser'
Nature and intelligence exist solely to serve the seer's true purpose - emancipation.
Intelligence exists to serve as the seer's agent, to liberate the consciousness from avidya.
If the sadhaka slackens in his sadhana and becomes inattentive, the senses agitate the seer and he is again hitched within the pleasures of the senses. This study of mind and investigation through intelligence is the innermost pursuit - antaratma sadhana.
This sutra conveys that consciousness, the essence of nature, which is cognisable, exists for the sake of the seer who sees for the sake of seeing.