I believe your question is not how you get a sine wave but how you get a sine wave that swings positive and negative about zero?

If that is your question, it's because transformers only transform ac voltages, not dc biases. So, the secondary output is completely independent of whatever the input was, assuming you are able to live with any dc current flowing in the primary.

Is that your question?

The output of this circuit will give an alternating current - but the voltage is floating - it will not be 'positive' or 'negative' until it is referenced to 'something'.

To get an output with excursions above and below earth potential, one of the transformer outputs should be earthed.

"is the purpose of the LC circuit use to shape the squarewave to a sine wave and has nothing to do with the positive and negative cycle thing?" - yes.

Like JohnDG said, the output (Secondary) side of the T2 is isolated or not tied to any reference. If you draw a tap from the center of that secondary winding and tie it to the ground, only then you can see the ac wave form cycle from somewhere +115 to zero and then to -115, and so forth.