Turaev's book assumes familiarity with basic 3-dimensional geometric topology and especially Dehn surgery presentations of 3-manifolds. If you want to understand all the details in Tureav's book, then I strongly recommend first reading Rolfsen's "Knots and Links", or some similar text.

It's hard to explain without pictures, but briefly: Start with $S^2\times I$. Remove a regular neighborhood of the arcs (not the loops) of the tangle in Figure 2.4. Do Dehn surgery along the (framed) loops. The boundary of the resulting 3-manifold is the union of a "vertical" annulus for each straight arc of the tangle and "upper" and "lower" surface. The upper surface contains $S^2\times \{1\}$ (minus some disks) and an annulus for each curved arc of the tangle. Call this surface $Y$. Then the 3-manifold, after Dehn surgery, is homeomorphic to $Y\times I$. Turaev's Figure 2.5 shows a 3-punctured disk which, after Dehn surgery, becomes an instance of curve$\times I$ inside $Y\times I$.

If the above explanation makes no sense to you then you should read Rolfsen.