This is a basic question (I think). I am trying to grasp the idea behind this example, where we define a "logistic function" and use that to work towards the maximum likelihood estimate (MLE).
We have a sample of $n$ people, with recorded data $(x_1,y_1),\ldots,(x_n,y_n)$, where $x_i$ is the BMI and $y_i\in\{0,1\}$ indicates whether the individual has some disease. We model the $x_i$ as non-random, and the $y_i$ as absolute values of random variables $Y_i$, where $$Y_i\sim\mathrm{Bernoulli}\big( f(x_i;\beta_0,\beta_1)\big).$$ Here, $f$ is a logistic function $$f(x;\beta_0,\beta_1)=\frac{\exp(\beta_0+\beta_1x)}{1+\exp(\beta_0+\beta_1x)}.$$
We then observe that if $\beta_0+\beta_1x$ is "very positive" then $Y_i$ is "likely to be $1$", and if $\beta_0+\beta_1x$ is "very negative" then $Y_i$ is "likely to be $0$".
My question is, what is a "logistic function" in this context, and why is it useful? I have a feeling that $f$ is supposed to describe the way BMI affects the probability of this disease. However, assuming that the relationship is in the form $\frac{\exp\cdot}{1+\exp\cdot}$, and depends linearly on precisely two parameters $\beta_0$ and $\beta_1$, seems like a huge leap, and one that needs to be justified somehow.
The rest of the example is copied below, just in case it is relevant to my question (I do not think it is).
Observe that $$1-f(x_i;\beta_0,\beta_1)=1/(1+\exp(\beta_0+\beta_1x_i),$$ so $$\frac{P_{\beta_0,\beta_1}(Y_i=1)}{P_{\beta_0,\beta_1}(Y_i=0)}=\frac{f(x;\beta_0,\beta_1)}{1-f(x;\beta_0,\beta_1)}=\exp(\beta_0+\beta_1x_i).$$ The likelihood function is \begin{align} L(\beta_0,\beta_1) &= P_{\beta_0,\beta_1}(Y_1=y_1,\ldots,Y_n=y_n)\\ &=\prod_i P_{\beta_0,\beta_1}(Y_i=y_i)\\ &=\prod_i f(x_i;\beta_0,\beta_1)^{y_i}\big(1-f(x_i;\beta_0,\beta_1)\big)^{1-y_i}. \end{align} In this case, the MLE has to be computed numerically. If $\hat{\beta_1}$ is large and positive, the model is appropriate; then if you recorded $x_{n+1}$ being large, this indicates a high probability that $Y_{n+1}$ is $1$. This does not imply a causal relationship.