「主部」とは、文の中の動作をする人・物で、日本語に直した場合に「～は」「～が」になる部分です。

「」、・、「～」「～」。

The 'subject' is the person or thing performing the action in the sentence, corresponding to what becomes '~wa' or '~ga' when translated to Japanese.

Literal

''Subject'' [quotative-とは], sentence [possessive-の] inside [possessive-の] action [object-を] do person/thing [copula-で], Japanese [locative-に] translate [case-場合] [locative-に] ''~wa'' ''~ga'' [locative-に] become part is.

A grammar textbook definition, apparently teaching Japanese speakers the concept of 'subject' using their own language as the reference. 主部 is the Japanese grammar term for the subject-portion of a sentence. 人・物 (person or thing) is the broad definition. The 場合 clause sets up a conditional framing: 'in the case of translating to Japanese'. The final part identifies the subject with what gets marked by は or が.