Could you please explain in a very simple way why I need a transistor? Why can’t I use my Arduino to switch the motor?
Because for an almost certainty an Arduino digital pin cannot supply enough current, and even if it could, that would only be at 5V. So while an Arduino digital pin can provide up to say 20mA at 5V for an LED, and switch it off and on, even tiny vibrator motors are beyond that capability. This one, for example, is 70mA: almost 4x too much for an i/o pin.
So you need to provide the right voltage (which might be 5V but just as likely not) with the right current capability, and use the Arduino digital pin to in turn control a transistor to switch that other supply.
It would look something like this:
Problem is, when you do that, you can only have the polarity one way; to switch the polarity to get the motor changing direction requires a so-called h-bridge. It’s quite complex to roll your own h-bridge, hence the availability of those like in the Pololu list.