When points have higher weight, they matter more in the loss function, and so the classifier will focus more on predicting them well at the expense of lower-weighted points.
In boosting, misclassified points are given higher weights so that subsequent classifiers are incentivised to learn patterns which explain points that have not been explained well by previous classifiers. Or, another way of putting it - subsequent classifiers are incentivised to correct mistakes of previous classifiers by the use of higher weights on misclassified points.