They are accelerating charged particles Accelerating charged particles make electromagnetic waves. Light comes out of magnetic poles like lights from a lighthouse. If you are in the way of the beam, you see the beams regularly. So you see pulses periodically. For a new supernova, this would be perhaps 1000 times per second.
Particles thrown out by the rapidly rotating core can heat the surrounding region, making it glow.
Energy thrown away by rotating neutron star can slow it down, so most pulsars pulse more like once per second instead of one-thousand times per second. Because stars are very heavy and rotating fast, they rotate extremely regularly. Rotation can be more regular than our best atomic clocks. There is a double pulsar. Timing is so perfect we can completely understand the orbit, and even see effects of Einstein's General Relativity.