"I'd say scrape a bit of the laquer off that covers the PCB and track..."

The PCB = The "blue ring"/board, correct?
The track = ?:

PCB = Printed Circuit Board, the board with copper traces and components
The track = The copper traces/tracks connecting the legs of components

Between the blue dots are darker green patterns. Beneath the greenish laquer there's a copper layer. Where the green is darker there is copper making the connections), where it's lighter the copper has been removed (forming the connections).

Where the blue dots are looks like "vias" (connections going through to another set of copper tracks on the other side of the board). I doubt that the card has more than 2 layers (any additional ones would be invisible as they're buried inside the board, but might be detectable with a strong light behind the board...)

What does the other side look like? I'd expect bare copper tracks/pads for the buttons of the remote to connect across if the remote uses a conductive pad (or does it use small mechanical switches?)...

You need bare copper to solder to - if you don't want to solder to the bare pads on the other side (if there is such) you need to follow the copper tracks from the pads of the buttons you want to "remote activate" (I'd guess VolUp/Down, FForward/Reverse and maybe Menu and Cancel) and then carefully scrape away the laquer layer without cutting through the copper layer and then solder the wire onto that. You might be able to heat the vias and insert the wires there, using the premade hole (depends on how the via was made though)

Glueing down the wires to relieve pulling stress on the solder connection itself is a good idea. If there's an unused part of the board along an edge you might drill a hole there and run the wires through and make a knot around strip separating the hole and the edge...

I'd use a temperature controlled soldering iron on fairly low heat, with a fine tip. Prolonged/excessive heat can make the traces separate from the board.

/Michael