Most dSLRS use focal plane shutters with two curtains. The first curtain "opens" the shutter, and the second one "closes" it. The size of the gap between the two shutters determines the shutter speed. The smaller the gap, the faster your shutter speed. When you reach your maximum sync speed (usually around 1/200s, for most cameras), that's the fastest shutter speed where the whole sensor is uncovered by the gap between the curtains. Once you go faster than that, the gap between the curtains is smaller than the sensor.
The light pulse from a flash, however, is likely to be much faster than your shutter speed. And if you use a faster shutter speed, then either the top or bottom (or both) of the sensor will be covered by a curtain while the flash burst goes off. So you'll get dark bars at the top or bottom (or both) of the frame. Since most people don't want this, the camera will automatically limit the shutter speed to the maximum sync speed with flash.
There is a feature called high-speed sync (HSS) or focal plane (FP) flash, which can get the shutter synchronization and multiple pulses from the flash so that the entire sensor will be evenly illuminated for the length of the exposure, but this requires more sophisticated communication between the flash and the camera, and most built-in "pop-up" flashes cannot perform HSS. And you typically need a TTL-capable hotshoe flash to do this (although not all TTL-compatible flashes can do HSS, either. OEM (same brand as your camera) flashes are your best bet). Be aware, however, that HSS will lower the effective power and light output of the flash by roughly two stops.
See: Neil van Niekerk's tutorial on HSS, for more detailed information.