As he kept talking, he gradually got more and more excited.
Literal
Talking [while-ているうちに], he [topic-は] gradually excited-became [toward speaker-てきた].
~ているうちに ('while doing, during the course of') marks a timeframe in which a change occurs. だんだん ('gradually') and ~てきた both point at the same progressive shift. ~てくる here signals a change emerging over time, visible to the observer.