How does the equivalence principle lead us to suspect that spacetime might be curved?

To understand Einstein's thought processes,

imagine yourself in the sealed box, being accelerated through interplanetary

space at 9.8m/s^2. You grab the flashlight that you keep on the bedside

table and shine a beam of light perpendicular to the acceleration vector. Since the box is accelerating upward, the path of the light beam

will appear to you to be bent downward, as the floor of the box rushes up

to meet the photons. However, thanks to the equivalence principle, we can

replace the accelerated box with a stationary box experiencing a constant gravitational acceleration. Since there's no way to distinguish between these

two cases, we are led to the conclusion that the paths of photons will be

curved downward in the presence of a gravitational field. Gravity affects

photons, Einstein concluded, even though they have no mass. Contemplating  the curved path of the light beam.

Fermat's principle, which states that

light travels between two points along a path which minimizes the travel time  required. In a vacuum, where the speed of light is constant, this translates  into the requirement that light takes the shortest path between two points.  In Euclidean, or flat, space, the shortest path between two points is a straight  line. However, in the presence of gravity, the path taken by light is not a  straight line. Thus, Einstein concluded, space is not Euclidean.  The presence of mass, in Einstein's view, causes space to be curved.

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