Way-of-speaking [by means of-で] he [copula-だ] [quotative-と] understand.
話し方 uses the suffix ~方 (かた, 'way of doing') attached to the verb stem — a productive pattern: 食べ方 (way of eating), 書き方 (way of writing), 読み方 (way of reading/reading of a kanji). で marks the means of recognition.