Why does falling water become less and less focused as it falls down?

I noticed that the beam of water coming through it became less and less focused as it falls down more.

You are probably referring to the waterflow becoming turbulent. The flow rate of water is constant especially when it is from a water purifier. But this phenomenon is not due to the fluctuations in the flow rate of water.

Since the flow rate of water is constant, its velocity too is constant. Therefore the flow of water is streamlined. But as it falls down, its velocity increases due to gravity and hence the linear dimension too. These increase in velocity($u$) and linear dimension($L$) causes the increase in the value of Reynolds number($Re$). Reynolds number is used to predict the type of flow of a fluid. Higher is the value of Reynolds number, higher is the probability that the fluid will be under turbulent flow.

The Reynolds number can be calculated by the following formula:
$$Re=frac{ρuL}{μ}$$
Here,
$ρ$ is the density of the fluid
$u$ is the velocity of the fluid
$L$ is the linear dimension
$μ$ is the dynamic viscosity of the fluid.

So you see, how the change in velocity and linear dimension affects the flow of water:)