In Witten's “QFT and Jones Polynomials” paper, page 383, it states that: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) by repeated surgeries on knots.

What are the methods to show this? In the simplest intuitive level?

I understand this may be a relevant post, but I hope there are better illuminations. Thanks.

p.s. I am a QM/QFT theorist trying to understand the topology better.