Why does water flowing from a tap become thinner as it flows down

Is it because the velocity of water increases as it flows down which results in a decrease in cross sectional area?

I believe you're right. We can also add that in the usual case of flow conservation, that water flow is constrained to "stick in one piece" by the tube inside of which it is flowing. In the case of the tap, the water flow binds together because of surface tension forces. But when the flow becomes to thin, an instability occurs as you can see it loses its circular shape.

Yes, you are right. The conservation of mass (for incompressible fluids, as water is) states that

$$ A_1 V_1 = A_2 V_2 $$

where $A$ is the area of the stream and $V$ the velocity of the fluid. Hence, as water accelerates due to gravity, $V_2>V_1$, so

$$ A_2 = frac{V_1}{V_2}A_1 rightarrow A_2<A_1 $$ and the stream becomes thinner as the water flows down.

it is because the velocity of water inreases as it flows down which results in a decrease in cross sectional area

There are two reasons for this, as follows.

First, surface tension tends to pull the jet of water together as it falls down, reducing its cross-section.

Second, the center of the jet is initially moving faster than the outer skin of the jet, and the flow is laminar. This also tends to urge the jet together rather than allowing it to spread out.

Note that this is true only for the case of a small jet that is moving relatively slowly, so that any aerodynamic effects are negligible.

Note also that once the jet has narrowed significantly, it becomes susceptible to a condition known as rayleigh instability, in which a tiny perturbation will cause the jet to separate into individual sphere-shaped droplets with a characteristic spacing or "frequency".