"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。"
Someone could explain better than me, but it is also used as a connection word. You can search for it, some grammar really helps here.
I don't understand why 很 is necessary when the adverb 最近 is already there? I thought the only rule was that Chinese forbids "bare adjectives" at the end of a sentence like 忙, so 很 seems redundant?