I agree it's still fun.
Kalman filters work by the assumption that everything is distributed by a gaussian. However the given example shows a case where we have very non-gaussian distributions, namely the input to the system is estimates of the distance of the lawn mower to the towers, which means it's a sort of annular distribution with a big whole in the middle (these are shown in the visualization). Contrast this to a gaussian which is a solid blob: a gaussian would be a poor approximation to the information provided by a distance to a point.
But without the assumption that the distribution is gaussian, the math is intractable, in particular how do you even represent an arbitrary probability distribution over possible positions? One option is to discretize space and give a 'heat map' of probabilities. This works but limits spatial resolution and grows to be a huge amount of work and memory for higher dimensional state spaces(1). The small dots are the alternative approach and are the eponymous 'particles'. Here we estimate the probability distribution by a collection of possible states, each being one point, and each state is weighted by how probable it is. Then a new piece of information applied (another round of distance data from the three points) to each of those points. This updates the probability of each point being 'correct'. Then a new batch of points is randomly selected by adding some noise to the current batch and preferentially choosing the higher probability points. This is why they jump around at each time step.
The advantage here is that most possible states are ridiculously unlikely but particles tend to group around the likely points, so we spend our time looking near the likely places and not around the unlikely places. If you discretized space, the vast bulk of the space will just be epsilon probability and doing you no good. And the particles have arbitrary precision without needing to increase the amount of points used. Plus we can work with non gaussian distributions. Otherwise it's 'just' another way of approximating a Bayesian update but under different assumptions and tradeoffs compared to a kalman filter. If you've got nearly gaussian distributions kalman filters are simpler and faster and more memory efficient, so they're very popular for embedded systems.
(1) Actually we're probably dealing with a 4 dimensional state space already making a discretized approach essentially already impossible. There are two space dimensions plus two for the velocity of the mower.