It suggests a large quantity or amount, often exceeding what is expected or necessary. Here are some examples:
* "She apologized profusely for her mistake." (She apologized a lot, showing deep regret)
* "The garden was blooming profusely with colorful flowers." (There were a lot of flowers, covering the garden)
* "He sweated profusely during the workout." (He sweated a lot because he was working hard)
Think of it as overflowing or abundant.