Why did Gothic builders switch to the pointed arch?

Gothic builders chose the pointed arch because it pushes less sideways than a round arch of the same span and load. Medieval European builders recognised this reduction of thrust quickly. Less sideways push means the walls holding up a stone ceiling can be thinner and taller, with room for large windows.

The shape itself is simple. A pointed arch has a pointed crown where its two curving sides meet at a sharp angle, and its form comes from the intersection of two circles, which is why it is also called a two-centred arch. It was not a medieval French invention: crude pointed arches have been found at the Bronze Age site of Nippur, dated earlier than 2700 BC, and the arch appeared early in Islamic architecture, in Umayyad and Abbasid buildings of the late 7th to 9th centuries.

What changed in the 12th century was how France and England used it, as an important structural element together with the rib vault and later the flying buttress. In the old barrel vault, a round arch pressed straight down on walls that had to be very thick, with few windows. With ribs of pointed arches, the weight was spread outwards and down to rows of pillars, so walls could be thinner and higher.

Builders also had a practical reason to like it. At Durham Cathedral, pointed arches let arches of different spans rise to the same height. And a common Gothic version, based on an equilateral triangle, could be drawn on stone with a cord and a marker, so arch stones could be cut at the quarry in quantity and still fit on site.