Think in the sense of power which is the product of voltage, current and power factor. When you connect the load at secondary,  current flow starts and power increases with increasing load but the power at the primary and secondary remains same. so primary power also increases. Now it depends upon the usage of transformer that it is step up or step down, You can guess what will happen at primary.  i.e. it is the matter of current or power handling capability at constant voltage that sets the limits. Voltage levels have been established and transformed is fully energized then it is only the current and losses which varies with load.  

The previous answer is perfect. Another answer is the following. When the current flow starts in the secondary the common magnetic flux increases, the electromotive force in the primary coil increases in the opposite direction to the voltage at the primary leads and the current at the primary increases to compensate the flux density generated by the secondary coil.

•  In present day transformer(M2H,N3H,M4 grad CRGO) saturation flux density is 1.9T. The working flux density is around 1.69T, that is saturation flux equal to 1.125 peak value of flux.
• Transformer works below, saturation in linear region of B-H curve and produce secondary pure(without harmonics) voltage depending on turn ratio(stapup or stapdown).
• Secondary current  increased or decreased, produces change in secondary ( increases or decrease)  amperturns(ATorMMFor(H)) .To balance this increased  or decrease AT, and to keep the fluxdensity(B -working) in linear as per magnetizing (I0) current,for required secondary voltage, the primary AT changes, and as primary turns(N1) is constant, the changes acures accordingly in current (opposite MMF), and thus the primary current increases or decreases. For detailed understanding refer my paper,'Laboratory methods to generate inrush current pattern...........power transformer'. 

The answer lies in the Phasor(Vector) Diagram of the Transformer on Load.

Transformer is a Device that transfers Electrical Power from one Circuit to another circuit without actual Electrical connection between the two circuits, by means of Mutual Induction though the magnetic circuit.

The Magnitude of the  Emf induced in the Secondary is decided by the Turns Ratio and is in Phase Opposition to the Voltage across the Primary.

When the Secondary is connected to the load it draws current from secondary voltage source(EMF) and sets up a MMF that is opposite to the one set up by the Primary.According to  Lenz's law,the Primary sets up an MMF to neutralize the Secondary MMF and does it by drawing a current from the main supply to the primary which is equal and opposite to the Secondary Current referred to the Primary as decided by the Turns Ratio..

Thus the change in MMF is the one that informs the Primary that the Secondary is loaded.

P.S.

When an ac voltage is connected across the primary terminals of a transformer, an alternating current will flow and self induce a voltage in the primary coil which is opposite and nearly equal to the applied voltage. The difference between these two voltages allows just enough current in the primary to magnetize its core. This is called the exciting, or magnetizing, current. The magnetic field caused by this exciting current cuts across the secondary coil and induces a voltage by mutual induction. If a load is connected across the secondary coil, the load current flowing through the secondary coil will produce a magnetic field which will tend to neutralize the magnetic field produced by the primary current. This will reduce the self induced (opposition) voltage in the primary coil and allow more primary current to flow. The primary current increases as the secondary load current increases, and decreases as the secondary load current decreases. When the secondary load is removed, the primary current is again reduced to the small exciting current sufficient only to magnetize the iron core of the transformer.

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