Relationship between log-odds and weighted sums in Logistic Regression
Ordinary regression, Logistic regression, Poisson regression are all examples of Generalized Linear Models (GLM).

A GLM can be broken down into 3 components:


A random component: Target Variable $Y_i$ which is assumed to follow a distribution with $E(Y_i)$ and $Var(Y_i)$ 
A systematic component: A linear combination of the features variables known as the linear predictor; $\gamma_i= \beta_{0} + \beta_1{x_1} \dots + \beta_n{x_n}$
Link Component: The link function, $g(\cdot)$, connects the random component to the systematic component. If $E(Y_i) = u_{i}$, then the link function connects $\mu_{i}$ to $\gamma_{i}$ through some function $g(\cdot)$, and we then have that
$g(\mu_i)= \beta_{0} + \beta_1{x_1} \dots + \beta_n{x_n}$. Moreover, $E(Y_i) = \mu_i = g^{-1}(\beta_{0} + \beta_1{x_1} \dots + \beta_n{x_n})$


The link function for linear regression is the identify link: $g(\mu_i) = \mu_i$

The link function for logistic regression is the logit link: $g(\mu) = log(\frac{\mu}{1-\mu})$. The logit link is used when $\mu$ takes values between 0 and 1 since the function maps to -infinity to infinity.

There are a lot details which I have skimmed over on purpose, GLM is a rich subject. I would recommend reading more about it, here are some references below:

References:

Categorical Data Analysis by Alan Agresti

An Introduction to Categorical Data Analysis

Hope this helps!