What exactly is a voltage divider? How does it work (for a beginner)
If you have 5 volts across a 5 kohm resistor then 1 mA flows through the resistor. That's ohms law: I = V/R. If instead of one 5 kohm resistor you had 2 series resistors of 2.5 kohm each you still get 1 mA flowing but the voltage across each resistor is only 2.5 volts because, rearranging ohms law to this: V = IR you get 1 mA x 2500 ohms. That is the very basic explanation of a voltage divider.
The tutorial said that a wire has to be connected from the Vcc to an
analogue pin on the Arduino for the readings. But how can a wire from
the Vcc send analogue signals about the soil's resistance? Doesn't it
just supply power?
Look closely at the picture in your question and note the resistor (appears to be 57 kohm): -
So, there is a resistor in series with the power feed to your moisture sensor probes and, if the soil had a resistance of 57 kohm, then the voltage across the soil probe would be half of the supply voltage (Vcc).
The tutorial mentioned a capacitor between the Ground and the signal
wire for filtering noise. What is this noise? How can the capacitor
filter it? Why does it need to be filtered?
Well, your soil will "pick up" all sorts of AC signals especially 50/60 Hz and this will become superimposed on the DC level that represents your soil's resistance. That AC signal may be large enough to make taking DC measurements a bit tricky so the capacitor will flatten out that AC signal leaving mainly just the DC signal you want to measure.