This is a Youtube video and in this part the speaker describes the early French colonies. As you see in the transcript at the bottom of the picture, he says "the western half of North America", but in the picture he shows (in the blue) the eastern part Is it a mistake?
The slide is talking about the area of land that French owned. The blue part is "in the eastern half" of the continent, but it is not "the eastern half" of that continent. As you can see from the map, the part east of the blue part is not blue: that area was British and Spanish, not French. All of that area, from the blue eastward to the ocean, was already settled by Europeans -- just not French ones.
So the writer probably means the French owned the western half of the "settled by Europeans" portion of the North American continent (north of Mexico).