Where the two voltages come from
The secondary is a single winding with a connection brought out from its midpoint. Across the whole winding, X1 to X3, you have 240 volts. From either end to the midpoint you have 120 volts. It is one winding giving two voltages, not two separate windings.
The centre tap is the neutral
That midpoint, X2, is bonded to ground at the transformer and becomes the grounded neutral of the service. Every 120 volt circuit in a house runs between one hot leg and that neutral. Every 240 volt circuit, a range or a dryer, runs between the two hot legs and frequently does not use the neutral at all.
Why the current is so high
Single phase current is kVA times 1000 divided by the voltage, with no square root of three. A 75 kVA single phase transformer delivers about 313 amps at 240 volts, where a 75 kVA three phase unit at 480 volts delivers about 90. Same kVA, very different conductors.
Balancing the legs
Because the 120 volt loads split across two legs, an unbalanced house puts current on the neutral equal to the difference between them. Panels are laid out to alternate 120 volt circuits between legs for this reason. A badly balanced panel is not dangerous but it wastes capacity.
Common questions
How is a single phase transformer wired
One high voltage winding on the primary, and a secondary winding with a centre tap brought out. 240 volts appears across the full secondary and 120 volts from either end to the centre tap, which is bonded to ground and becomes the neutral.
Why does a single phase transformer give both 120 and 240 volts
Because the secondary is one winding with a connection at its midpoint. The full winding gives 240 volts, and either half gives 120.
Is the centre tap the neutral
Yes. The centre tap is bonded to ground at the transformer and becomes the grounded neutral of the service. All 120 volt circuits run between a hot leg and that neutral.