How do I determine this logistic growth model formula?
The growth of the yeast can be studied with a Logistic model (i.e. differential equation) as follows:
$$
\frac{dX}{dt}=\mu X \left(1 - \frac{X}{X_{max}} \right)
$$
This is an ordinary differential equation that tells you how the population of yeast is changing with time (in fact is telling you how the concentration of Yeast $X$ changes with time).
The two parameters in the equation are the specific growth rate $\mu$ and $X_{max}$ the carrying capacity following the Verlhust model.
We could also write the equation following your notation:
$$
\frac{dN}{dt}=r N \left(1 - \frac{N}{K} \right)
$$
where $r (\mu)$ is the specific growth rate, $K (X_{max})$ is the carrying capacity, and $N$ is the number of elements in the population.
Note that this is a dynamic model that you need to solve (i.e. integrate the differential equation) to be able to compare with your experimental data. This model tells you how any population of this time behaves not only your Yeast in the mentioned experiment.
The solution of this model is the following Logistic equation:
$$
N(t) = \frac{K}{1+\frac{K-N_0}{N_0}e^{-rt}}
$$
Where $N_0$ is the initial number of Yeast cells.
If you fit this equation (i.e. with least-squares) you get the following values for your experiment (with their 95% confidence intervals):
K =       603.6  (594.3, 612.9)
N0 =      2.108  (1.179, 3.036)
r =      0.6729  (0.6196, 0.7262)
And the following curve.

$r$ is the growth rate (in $hr^{-1}$) that is related with the doubling time by the following formula:
$$
t_d = \frac{ln 2}{r} = 1.0301 hr
$$
Check also this answer about growth models