This is the 3023rd most frequent Russian word.
завоевать
To conquer or win.
The word "завоевать" in this sentence is used metaphorically to mean "earn" or "gain" recognition.
Он мечтает завоевать признание аудитории.
He dreams of earning the recognition of the audience.
Here, "завоевать" is used in its literal sense, meaning "to conquer" or "to win control over."
Генерал планировал завоевать важный стратегический город.
The general planned to conquer an important strategic city.
In this sentence, "завоевать" means to "win" or "achieve" with effort, emphasizing attaining respect or admiration.