The mother is looking after his care.

Literal

Mother [subject-が], his nursing [at-に] is-attending-[ongoing-てる]

看護にあたる = 'to attend to nursing/care' — あたる (to hit/face) takes に to mean 'to take on / deal with.' あたってる is the casual contraction of あたっている (い dropped). The comma after が sets off the subject.