Representing belief as many weighted samples
A particle filter represents the belief distribution concretely, as a large cloud of individual guesses called particles, each one a candidate position, and often orientation, that the robot might actually be at. A typical implementation might maintain hundreds or thousands of particles scattered across the plausible space. Instead of one number describing the robot's position, the filter tracks an entire population of hypotheses simultaneously, with denser clusters of particles indicating positions the filter considers more likely.
Each particle carries a weight reflecting how well it currently explains the sensor data. During the update step, every particle simulates what a sensor reading would look like if the robot really were at that particle's position, given the known map, and compares that simulated reading against the sensor reading the robot actually received. A particle predicting a wall five meters ahead when the real sensor sees open space gets a low weight. A particle whose predicted reading closely matches reality gets a high weight.