Well, I see how I should have been more explicit in how to connect the MP3 module to the amplifier module. 
Only 1-wire connects the speaker output of the MP3 player to the audio input of the amplifier module.
The "why" has to do with grounding and a truly techno-geeky concept known as BTL (bridge-tied-load). So all you need to do at the audio amp input is connect the "L" pad to one of the two speaker output pads on the MP3 player. It doesn't matter which pad you choose, but to keep things consistent note that I used the pad closest to the corner of the board - the orange wire in my picture.
And, yes, connecting two wires indeed explains why the magic smoke was released. You can ignore the elaboration but here's the technical mumbo-jumbo if you'd like to glaze your eyes. Modern DC powered audio devices (smartphones, tablets, laptops, etc.) use so-called BTL audio amplifiers. Neither of the 2 output wires to the speaker in a BTL amplifier are "ground". Simply stated, the 2 wires are mirror images (one positive, one negative) and is essentially a method to "double" the audio output to a speaker. The middle pad on the 3-wire input of the audio amp module is "ground". So what happened is you connected one of the BTL outputs directly to ground which amounts to a short-circuit to the poor audio amp on the MP3 module.
Again, I apologize for not explicitly warning you that only 1 wire should be connected to transmit audio from the MP3 player to the audio amp.
