Question about Logistic Regression
It is just a matter of modelling. 

Note that $w \cdot x = \sum_{i=1}^n w_i x_i$.

For example in linear regression (ignore intercept term for now, we can always insert the intercept term by including $1$ as a feature), we try to fit a model of the type of $y(x) = \sum_{i=1}^n w_ix_i$, however, the problem with this when the dependent variable is that $y$ can take values beyond $0$ and $1$. 

Hence, we consider another model $F(p) = \sum_{i=1}^n w_ix_i$ where $F(p)$ can take any real value and $F$ is a function which takes value from $(0,1)$ and be able to map it to $\mathbb{R}$. $F(p)$ can be chosen to be $\log \left( \frac{p}{1-p} \right)$, which is an increasing function which map $(0,1)$ to $\mathbb{R}$.