Joy-IT RS232 Shield and RS232 voltage levels
Yes, that shield will handle the level shifting for you. If you look closely at the images of the shield, you see the MAX232 chip, which is a commonly used chip for getting UART (which is the Serial interface of the Arduino) to speak over RS232.
And yes, RS232 uses different voltage levels. The UART on a microcontroller usually uses the voltages 0V and 5V (or 3.3V depending on  it's supply voltage), which is also called TTL levels (Transistor-Transistor Logic), while RS232 mostly works with +-10V (though different voltages are also inside the standard). You see, that this is a symmetrical voltage around ground (0V).