When telling a story with a scatter plot, chances are you want to tell something about some specific data points, like the data points with very high or low values, or the data point representing your country among all EU member states, for example. With labels, you can highlight these data points, draw attention to them and make it clear what each of the data points represent.
Because of the amount of data points and clusters in the data, it might not be feasible to label every data point on a chart. But this is probably not needed for the point you are trying to make with the scatter plot. When a scatter plot is published online and is interactive, the chart can offer tooltips that reveal data about each data point when it is clicked or hovered over.
Hover over a dot to reveal its data. Source: Datawrapper
Labels could include the values for x and y for the labelled data points, or this information could be revealed through the tooltips.