To play music in Fire Emblem Echoes: Shadows of Valentia, you first need to start the game. Then, wait until the game prompts you to press a button. DO NOT PRESS THE BUTTON YET!
You’ll need the button on the bottom left of your 3DS (the one shaped like a +) to input a sort of code.
Press the button in this order when it tells you to: Up, Down, Left, Right, Up
Next, head over to Extras, and you’ll find the option for 'Music' (or 'muziek', depending on your language settings), where you can listen to all the music from the game.
The video I’ve included provides a clearer explanation of everything.