The linings are strips glued to the inside surfaces of the thin sides to make a wider glue joint with the top & back panels. I have made them two different ways so far: as flat strips bent to the same shape as the sides, and as (thicker) kerf-sawn strips. Either way is probably fine, kerf-sawn is more "normal". Pick a wood which is easy to bend.
Making kerf-sawn strips on the band saw:
Once you have the strips, glue them to the sides, top and bottom. Clothes pins are invaluable here. Try to get the strips even with the edge of the side, err on the high side (so you're sanding the lining instead of the side).
Once the glue dries, sand the top & back surfaces flat and smooth by hand, with a wide sanding block.
Next: Glue on the back
If you find any of this helpful I'd certainly appreciate a "tip" :-)