To answer your question it is necessary to explain how a transformer works. Also you have to understand the complex notation for currents and voltage. I am not going to explain it, if you don't understand it, please check some physics textbooks.
Both coils of the transformer are wound around the same core. Let us say that the primary goes around the core $N_1$ times and the secondary $N_2$ times.
Depending on the size of the core and the exact nature of the ferromagnetic substance in it the magnetic flux in the core caused by a current $I$ in a single loop will be $\Phi =\ell I$. Consider a current $I_1$ in the primary and a current $I_2$ in the secondary. For convenience I am counting the relative directions of the currents in opposite ways so the total flux in the core is
$$\Phi_{core}=\ell (N_1 I_1-N_2 I_2)$$
The total flux in the primary is $\Phi_1 = N_1\Phi_{core}$ because it wounds $N_1$ times, while the in the secondary it is $\Phi_2 =N_2 \Phi_{core}$
The calculations would be much more complicated if one takes into account Ohmic losses in the coils. Therefore I will suppose that the coils are perfect conductors, and ignore Ohmic losses in them. This of course is not exactly true, but it better be a good approximation, otherwise your transformer will cost you a lot of money in charges from the Electricity Company...
In the absence of Ohmic losses inside the transformer, the voltage difference between the endpoints of the primary and secondary are $-d\Phi_1/ dt$ and $-d\Phi_2/ dt$ respectively. In complex notation the time derivative, for an alternating current of frequency $\nu$ (for instance 50 periods per second) is obtained by multiplying by the complex number $-i\omega$ where $\omega=2\pi \nu$. To simplify notation I will call $L$ the quantity $2\pi \nu \ell$ but remember, here $L$ already takes into account $\omega$ and thus my notations differ from the usual ones.
With all these preliminaries the voltage at the primary is $iLN_1(N_1 I_1-N_2 I_2)$. I also assume your Electricity Company has enough power to guarantee that this voltage is indeed the promised one, namely $V$, be it 110V or 220V, whatever amount of electricity you use (assuming there are no losses in the wiring inside your apartment, which is not exactly true but better be also a good approximation.....)
So at the primary we have
$$V=iLN_1(N_1 I_1-N_2 I_2)$$
and therefore at the secondary the voltage is
$$iLN_2(N_1 I_1-N_2 I_2)=(N_2/N_1) V$$
If $N_2$ is larger than $N_1$ you have a voltage raising transformer.
Now what are you doing to do with this output voltage ? If you run an electric motor, things will be too complicated. Just assume you run
it through a pure Ohmic resistor, for instance a light bulb of resistance $R$.
Then the current $I_2$ in the secondary circuit will be
$$I_2=(N_2/N_1) V/R$$
Now we can calculate $I_1$. We have
$$iLN_2(N_1 I_1-N_2 I_2)=R I_2$$
Thus
$$iLN_2N_1 I_1=(iLN_2^2+R )I_2 \to I_1=(N_2/N_1-iR/(LN_1N_2)\ )I_2$$
The ratio $I_1/I_2$ is the complex number $N_2/N_1-iR/(LN_1N_2)$ !
In normal utilisation context, $R$ is small compared to $LN_2^2$ (remember that in my notations $\omega$ is already counted into the definition of $L$ which is not the usual notation). So the imaginary part is small, and the ratio $I_1/I_2$ is close to the real number $N_2/N_1$.
In the case of a voltage raising transformer, the current in the primary is larger than in the secondary by a complex number close to the real number $N_2/N_1$, which can be expressed as
In the case of a voltage raising transformer, the current in the secondary is smaller than in the primary by a complex number close to the real number $N_1/N_2$, in normal utilisation.
But what happens if $R$ is large ? Than the imaginary part of the ratio $I_1/I_2$ becomes larger than the real part ! But it does not mean that $I_1$ becomes larger, it means $I_2$ becomes smaller. In fact
$$I_1=(\ (N_2/N_1)^2/R-i/(LN_1^2)\ )V$$
If there is nothing plugged into the secondary, it amounts to an infinite $R$. $I_2$ is zero, of course, but not $I_1$ (unless you also unplug the primary). But then this current is purely imaginary.
What ! When nothing is plugged in the secondary, there still is a current in the secondary ? And the company is charging me for it ?
No. An imaginary current does not produce power. You are only charged for power, not current. The meter of the Electrical Company only measures the real part of the current, the power producing one, not the imaginary one. It is built that way on purpose.
It you consider the power, indeed in the resistor, you have voltage $(N_2/N_1)V$ and current $I_2$. The real part of $I_1$, the only power producing one is $(N_2/N_1)I_2$ and delivered at potential $V$. So the power is indeed the same.
But you are right. Just the argument of power would not allow you to compute the imaginary part of $I_1$, which is present even with nothing plugged into the secondary.
But nothing is perfect, and there are Ohmic losses everywhere, the current $I_1$ is never purely imaginary when the secondary is unplugged, and thus the Electricity Company will measure its real part and charge it to you. So to save money and help our planet, do unplug the primary when you do not use the secondary.
