Levi-Civita connection is usually defined as the unique connection which is torsion free and preserves metric.
Question Is there some intuitively transparent constructive way to define it (or corresponding parallel transport) ?
"intuitively transparent" is up to "good will" of the ones answering.
I remember the following construction but it is not intrinsic and it is not clear for me how to derive the formula for Christoffel symbols from it in transparent way. Nevertheless let me mention it.
Assume we have a submanifold in some Riemann manifold. To define the transport along the curve on a submanifold we can do infinitesemal translation in bigger manifold - the resulting vector may not be tangent to submanifold - so we will make a projection on the tangent space of the manifold.
In this way starting from standard metric on R^n we can derive parallel transport and hence Levi-Civita connection on a submanifold. Expressing result in terms of submanifold's metric may be considered as way to answer the question - but it seems to be very indirect.