Should I explain my poor performance during undergrad, due to mental illness, when applying for PhD?
You do not need to disclose the nature of your illness if you do not wish to.  It was your illness, and you get to decide how you handle it.

The short explanation you propose is fine.  You might want to specify that it was the first three years:


  My performance during the first three years of my undergraduate studies was not representative of my abilities, and was due to illness. However, I am glad to say that I have fully recovered, and as my transcripts show, I was able to overcome those early difficulties.