"达到活动的高潮" can be translated into English as "reach the climax of the event" or "hit the peak of the activity".
Both expressions convey the idea of the activity reaching its most exciting or intense moment. Here are some example sentences for context:
The concert reached its climax when the lead singer performed the encore.
（音乐会在主唱表演返场曲目时达到了高潮。）
The sports event hit the peak of excitement during the final match.
（体育赛事在决赛期间达到了最激动人心的时刻。）