Read Section 4.1 yourself. I will briefly summarize.

Keep remembering that knots are closed polygonal curves, there is nothing smooth here. No smooth transformations of the space, just shifting the verticies and edges of the polygon.

We have equivalence of knots defined by elementary moves.
We define equivalence of diagrams defined by Reidemeister moves.
Reidemester's theorem says that two knots are equivalent if and only if their diagrams are equivalent. 
The <= implication is clear. 
In =>, we have to prove that if you create (or shrink) some crazy large triangle that intersects many edges of the diagram, then you can actually replace that one big elementary move by a sequence of smaller elementary moves that cross only a limited amount of edges and can be identified with one of the Reidemeister moves.

Make sure that you understand that the moves Omega_1,2,3 (Fig. 4.1.1) are sequences of elementary moves. They are called Reidemeister moves. The theorem essentially says that these are the only important sequences of elementary moves.
Shortcut: R-move = a sequence of Reidemester moves.
Fig. 4.1.4 and 4.1.6 (i.e., Lemma 4.1.2) show some interesting R-moves.
Lemma 4.1.3 clarifies that if you make this crazy large triangle that intersects many edges of the diagram, then for every piece of the polygonal curve that intersects the triangle, it cannot happen that one intersection is above and the other one below the triangle edge. This is the place where you use some geometry.
The rest of the proof explains how to actually decopose the creation of the large triangle into Reidemeister moves, resulting in something like Fig. 4.1.13.
 
What about next week? 
A property is invariant with respect to knot equivalence, if and only if it is invariant to the three Reidemeister moves. This will be exploited in the rest of chapter 4.