How to make this strange embossed apple logo?

It’s all a bit easier to discuss if we start with horizontal lines, like so:

As Ilmari Karonen said, this is just a displacement map. You start with a 3D topography h(x,y), and then each point (x,y) in the result should use the line starting at y₀ = y − h(x,y).

With raster graphics, this can be rendered straight away, but to get clean vector lines we’d actually need to work the other way around: each line should be parameterised as

f_y0(x) = y₀ + h(x, f_y0(x))

The problem is that this is only an implicit defining equation, because the RHS contains y = f_y0(x). This could be solved with Newton-Raphson, but there’s a simpler way: just construct the lines going from left to right, and at each point go up or down as needed to fulfill the equation. Assuming continuity, this is always just a small update, so we can approximate h as constant there.

Implementation in Haskell:

type ℝ = Double

horizContourLine ::
     ((ℝ,ℝ) -> ℝ) -- ^ The topography/height function
  -> (ℝ,ℝ)        -- ^ x-interval on which to render the path
  -> ℝ            -- ^ Step size / resolution along the path
  -> ℝ            -- ^ Base-level y-coordinate of the line
  -> [(ℝ,ℝ)]      -- ^ Trail line
horizContourLine h (x₀,xe) δx y₀ = go (x₀,y₀)
 where go (x,y) = (x,yTgt)
            : if x<xe then go (x+δx, yTgt) 
                      else []
        where yTgt = y₀ + h (x,y)

Then, with a suitable image input like

we get contours like

Full code: https://gist.github.com/leftaroundabout/8b7075d25adecdf16806ea5d7a7ea2aa

Rotation to diagonal lines is left as an exercise.