I'm reading a text about statistics, but I don't understand why Poisson applies. (Note, this is not an assignment or anything like that.)

Why would X be Poisson distributed with that parameter theta?
The only Poisson that I could find reasonable is modelling X as Poisson distributed with parameter [itex]\lambda t[/itex] where [itex]\lambda[/itex] is the rate of multiplication (of the bacteria), and t is the incubation time (which is mentioned in the quote, but strangely enough does not affect the probability distribution in the above case).

The 'counting' refers to the number of bacteria colonies in each petri dish, with each colony presumed to have arisen from a single bacterium. The number of bacteria in each 1 ml sample actually has a binomial distribution, but if there are many bacteria present, a Poisson distribution is a good approximation.

In what sense is it binomially distributed? What is n and what is p?

And the quote above mentions an incubation time. Shouldn't the distribution account for this time? I.e. shouldn't the incubation time have an effect on the distribution?

A Poisson process is memoryless and is characterized by a fixed rate parameter such as a half-life or doubling time. Many natural processes obey this model at least for parts of a process. Bacterial populations tend to increase at a fixed rate unless or until some limiting factors (ie food supply) kick in.

There are a discrete number of bacteria in each 1 ml sample. The probability of finding 0, 1, 2 ... bacteria in any given sample has a binomial distribution. n is the total number of bacteria present in the solution, and taking 5 samples allows one to estimate p and thereby estimate the bacteria density. Since there are a large number of bacteria, one can use a Poisson distribution as a surrogate.

The incubation time is immaterial -- it's there only to allow the colonies enough time to grow large enough to be seen and counted. In this type of experiment, one generally expects to see a few to a few dozen colonies, each a couple of millimeters in size, and each presumed to have grown from a single bacterium. Keep in mind that it's the bacteria density in the orginal solution that is of interest.

So the probability of finding 1 bacterium in an infinitesimal fluid volume dv is (N/V)dv.

That's where the Poisson distribution approximation comes from. It is an approximation, because it assumes the bacteria are geometrical points and they are dispersed "evenly" through the fluid, so that a small enough volume dv will never contain more than one bacterium - i.e. the geometrical distribution of bacteria in the fluid doesn't contain any limit points.

The Poisson distribution holds even if the mean density is non-constant (but still deterministic) - the Poisson parameter here will be the integral of the mean density over the region.