Understanding binary logistic regression as a linear model I understand that binary logistic regression is applied to binary classification problems where the dependent variable $Y$ has only two possible outcomes. The independent variables are $x$. The result of logistic regression is assigning a probability $p$ to one of the two outcomes and a probability $p-1$ to the other possible outcome.
I am confused on how the linear combination of the independent variables $w_1 x_1 +w_2 x_2 +w_3 x_3 $, $log \frac {p} {1-p}$, the probability $p$, the logistic function $\frac {1}{1+e^{-x}}$ are connected to each other.
Can someone help me logically understand how these concepts go together so I can finally appreciate how logistic regression works?
Thank you!

This is the logistic regression model, where the log-odds are posited to change as a linear function of some predictors.
$$
\log\bigg(
\dfrac{p}{1-p}
\bigg) = X\beta
$$
$X\beta$ is the linear combination. You denote it as $w_1 x_1 +w_2 x_2 +w_3 x_3 $. A more traditional way to write it would use $\beta$ as the symbol for coefficients and would involve an intercept, so more like: $$X\beta = \beta_0 +\beta1x_1 + \beta_2x_2+\beta_3x_3$$
In order to solve for $p$, we must do some algebra.
$$
\log\bigg(
\dfrac{p}{1-p}
\bigg) = X\beta\implies\\
\dfrac{p}{1-p} = \exp(X\beta)\implies\\
p = (1 - p) \exp(X\beta)\implies\\
p = \exp(X\beta) - p \exp(X\beta)\implies\\
p+p\exp(X\beta) = \exp(X\beta)\implies\\
p(1 + \exp(X\beta)) = \exp(X\beta)\implies\\
p = \dfrac{\exp(X\beta)}{1 + \exp(X\beta)\implies}\\
p = \bigg(
\dfrac{1 + \exp(X\beta)}{\exp(X\beta)}
\bigg)^{-1}\implies\\
p =\bigg(
\dfrac{1}{\exp(X\beta)} + 1
\bigg)^{-1}\implies\\
p =\bigg(
\exp(-X\beta) + 1
\bigg)^{-1}\implies\\
p = \dfrac{1}{1 + \exp(-X\beta)}
$$
