How does a particle know how to behave?

I think this question makes hidden, inarticulated assumptions about reality. In physics, we make observations and then try to find models that match them. The models, though, belong only to us and exist in our heads and textbooks.

We perform the calculations required to make our predictions in our models. We cannot say whether nature makes similar calculations, and asking 'how' nature or particles perform calculations seems wrongheaded, to me. Jaynes called this the mind projection fallacy; you are projecting things that exist in your mind - calculations and models - to reality.

Responding to @knzhou, I would not, though, advocate answering every question about physics with the fact that 'physics is a bunch of models'. It is possible to construct models, answer questions about them and make calculations without projecting our calculations to reality.

The question 'how does water know to boil at 100 C?' would be better phrased 'how does our model explain the observation that water boils?'. We could answer that with @knzhou's answer: 'bubbles of steam finally have enough energy to expand against the water pressure.' This is indeed a valuable explanation.

One way to think about it is that a particle "sniffs out" its immediate surroundings and reacts to gradient: a trend like a declining potential in one direction.

Single-celled organisms do this. Plant orient toward the sun. A rock on an incline "senses" that it's center-of-mass is slight off from the point of contact with the ground. This is all loosely speaking, of course.

In physics, differential equation capture the same idea. $F=\frac{dV}{dx}=m \frac{d^2x}{dt^2}.$ Tiny differentials give marching orders to particles and charges and spacetime.

There is another mathematical formulation, with Lagrangians and actions, where a particle chooses the path that minimizes the action, as if the particle knows which path to take. Or Fermat's theorem where light takes the path of least time, as if the photon is intelligent enough to compare a lot of paths. This may look like particles "know how to behave." However, mathematically these theorems are equivalent to differential equations. After all, you can divide a path into many tiny segments, and then you're back to the differentials of differential equations.

So it's all a very local computation that a particle needs perform. We find similar ideas all over science. For example, in (artificial) intelligence, there's the Hebb rule: many neurons are connected in a big network, but when the network "learns" each neuron makes small adjustments to how strong its connection is to nearest neighbors only. But as a result, the entire network can learn to perform a complex computation.

Hope this makes it feel a little more clear.

How does a particle know how to behave?

Already the title is about metaphysics, a consciousness is attributed to the particle by the verb "know".

Physics is about modelling observational measurements with mathematical formulae which can predict future behavior. The "knowledge" is collective and comes from an accumulation of an enormous number of observations.

There are metaphysical models which attribute consciousness to particles. A particular one I read in my metaphysical era ( and was always careful to separate physics from metaphysics) is the "units of consciousness" model of Janer Roberts who was channeling Seth ( you cannot get more metaphysical than this :) ).

In this model all that exists is units of consciousness, which, like many dimensional cosmic strings exist from -infinity to + infinity, building up nature as we observe it. In that metaphysical frame, the question has an answer.

There is no answer within Physics theories and models.