A quick look at the schematic shows an issue with Arduino Leonardo Board, preventing out of the box operation.

Support for Arduino Leonardo is afflicted by having the SPI pins not routing the same pins as the UNO. This is similar to the Arduino Mega. Where as it appears it should simply work with additional jumpers, from the Leonardo’s ICSP port, which has the SPI pins to the MP3 shields equivalent SPI pins.

D13 to ICSP3 – SCK
D12 to ICSP1 – MISO
D11 to ICSP4 – MOSI
and remember to not use D10

Please let us know if this works? I think it should.