The first confusion is where are we even measuring the voltage across from? Voltage across the capacitor? Across the resistor? Is there some kind of convention we normally measure on?
We measure the voltage with respect to the ground symbol, unless the explicit phrase 'voltage across the component' is used. The diagram falls short by not showing a ground line running continuously from the input terminals to the output terminals.
There is another confusion though, those arrows are intended to show the current passing through the components, not the voltage across them.
The impedance ZC increases as we decrease frequency, so the voltage drop across the capacitor decreases when frequency is low.
No, the voltage drop across a capacitor increases when the frequency of the current through it decreases.
Doesn't that mean that the capacitor is letting through all the low frequency signals... Why is the lower picture showing us that all the high frequency signals are going through the capacitor?
All the high frequency current is passing through the capacitor, none is exiting the output terminal and passing through the load.
So shouldn't the capacitor be a high-frequency blocking element? Why is it known to be a DC blocking element then?
A capacitor shunted across two terminals blocks a high frequency voltage from appearing across them, the capacitor creates a low voltage across its terminals.
A capacitor in series with a signal line blocks the flow of low frequency and DC signals, by allowing a large voltage to appear across its terminals.
There is yet another shortcut that the teacher is taking when talking about 'signals'. A signal comprises both a voltage and a current, on a pair of wires, which forms a complete circuit when joined at the source end through the source, and the load end through the load. It's usually obvious in context whether one is operating on the current or the voltage of the signal with shunt or series components, once you have a clear mental picture of the situation. You've not got that clear mental picture yet. You'll get there, with questions like this.