"I have been busy recently."
One way to think of this: "很" functions like the verb "to be" in English when discussing the degree of a state. In this case, the state is "busyness" and the degree is "busy".
So, to say "I am busy" in English, you say "我很忙。" in Chinese.
"Recently" (最近) is an indicator of time and must follow the subject like any other indicator of time resulting in "我最近很忙。"