questions about logistic regression

In the following Linear Regression discussion I didn't understand a few things:

So my questions are:

1. In the third slide: What does this probability means $P\left(y_i|x_i\right)$ and accordingly what does it mean to maximize it ? Does it mean to maximize both $P\left(y_i=1|x_i\right)$ and $P\left(y_i=0|x_i\right)$, and as higher this probability, the more stable and rightful results we get, and accordingly the more correct weights $w^*$ we get ?

2. In the fourth slide I don't see the math, could anyone detail it ? How did we get that result ?