Can somebody explain what the bold italic sentence means and what's the relationship between $\vec{w}$ and $\delta$?
This value of delta is ½ the margin for our decision boundary. In Euclidean geometry, the shortest distance from a point to a hyperplane is one that is orthogonal to the hyperplane (perpendicular to the line in 2 dimensions). In the plot, delta equals the difference between vector A and vector B. We know that delta is in the opposite direction of our normal vector (marked in red). We can use this fact and first compute the pure direction of our normal vector by dividing it by its norm and then multiply it by delta to gives us a vector of length delta and pointing in the direction of the norm. We then subtract this value from A to arrive at point B on our hyperplane.
$B=A-\delta\dfrac{\vec{w}}{||\vec{w}||}$
Presumably $\delta$ is being measured from $A$ to $B$, so all that the highlighted sentence is saying is that the vector $B-A$ points in the opposite direction from $\vec w$, which is why you subtract instead of adding in the formula at the end.