unity - How does transvoxel algorithm work and how can I implement It?

After wasting over three days of researching on DuckDuckGo/Google, trying to understand transvoxel paper and existing implementation, I come here to clarify the subject for me and made it easier for the next person who wants to implement it.

Here is the transvoxel's homepage which explains well what the algorithm solves. So now I'm happy to find something that solves my current problem, so I would like to implement it to my code, but the website doesn't explain how to implement it or sharing code (except tables). So I was looking for existing implementation of the transvoxel algorithm:

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  Transvoxel (XNA): Looks great but this solution all-in-one (octree, caching, voxel, etc.) I really tried to understand the transvoxel class into this one, but no.

  
  


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  Transvoxel (OGRE3D): Looks great too but again, all in one. I tried too to understand, I translated the code from C++ to C#, but no again.

  
  

Finding for marching cube was so easy, one class copy&paste, one public function.

So I gave up the idea to use existing implementation. Why not implementing it myself? (Sadly I don't have the skill to fully understand what the paper said, for example the ambiguous case).

The first thing I didn't understand was if transvoxel is a modified marching cube or a seconds layer that modify the work of the marching cube. In the C++ code (second link of the list) the code uses a implementation of marching cube and transvoxelize each border, on the paper they talk about 19x19x19 voxel data on 16x16x16 chunk and about a modified version of marching cube.

So do you guys can explain me how the transvoxel algorithm works and how can I implement it on my existing project?

My existing project is an Unity3D project which uses marching cubes for an infinite world, LOD chunk are ready, the only one thing missing is fixing crack between different LOD level chunk.