Energy - the multiplication of volts and amps (watts) is transferred from the primary to the secondary via the magnetic path or circuit made up of the core (the laminations). It's a bit like someone filling a bucket at the beach with water from a small cup and someone else removing the water with a small cup.
When the secondary switch is open, the bucket gets filled and the person stops filling.
If there is a slight leak in the bucket, the person adds a little water every few minutes.
This is exactly like the transformer with the secondary switch open.
The incoming voltage very quickly fills the magnetic circuit with flux and this flux opposes the incoming flux so that current falls to a very small value.
This occurs on every positive and negative half-cycle.
When the switch is closed, the secondary circuit removes some of the flux and the incoming waveform delivers more current for a longer part of the half-cycle – in attempt to fully fill the magnetic circuit with energy - flux.
The magnetic circuit is a bit like this:
Imagine a bucket at the beach with a pipe halfway down the bucket and this small pipe going into a second bucket.
The first person fills the bucket and the second person empties the second bucket, but the small pipe between the two buckets limits the rate at which the water can be removed from the second bucket.
This is the limitation of the magnetic circuit.
It can only transfer a certain amount of energy and this limits the wattage of the transformer.