The reason why "my wife and I" is correct is that it is the subject of the sentence. A tip on situations like this is to try replacing the phrase with a single pronoun:

"I picked up a puppy" is correct.

"Me picked up a puppy" is incorrect.

"Me" is correct as the object of a sentence.

"The puppy licked me" is correct.

"The puppy licked I" is incorrect.

So "The puppy licked my wife and me" is correct.

How is it correct?

How is it correct?
1. My wife and I picked up a homeless puppy recently.   This sentence is correct.  Keep up the good work!  Rachael Alice 

2. Me and my wife picked up some a homeless puppy recently.