Concentration-time graphs in determining order of a reaction

How/why does a concentration-time graph determine the order of a reaction?

Considering the relevant rate equations
Zero order: rate = k
First order: rate = k[A]
Second order: rate = k[A]^2
You might recognise that the rate of a reaction is the gradient of a concentration-time graph.
For a zero-order reaction, the gradient is a constant and so you get a straight line.
If you have studied A level maths or equivalent (in particular the integration topics covered towards the end of the course), you will recognise that the rate of the reaction is a derivative with respect to time (in this case -d[A]/dt, the negative sign assigned because as time progresses [A] depletes) and so you can form differential equations to find the relationship between [A] and t for first and second order reactions. The results should be as follows:
First order: [A] = [A]0 exp(-kt)
Second order: [A] = [A]0 / (1 + kt[A]0)
Where [A]0 is just the initial concentration of substance A.
The result is you get completely different shapes of graph depending on the order of reaction with respect to your reactant.

Thank you, I understand now. But how does it work chemically? For example, why does the concentration of a reactant with zero order change linearly over time?

Put simply, the answer is that it’s purely a mathematical coincidence - you get a differential equation of the form d[A]/dt = -k and it just so happens to be solved by the equation of a straight line, [A] = [A]0 - kt.
If you want a more intellectually satisfying answer, I suppose you could instead inquire as to why reactants can be assigned orders of 0, 1 or 2 (in simple cases). For that, you need to understand reaction mechanisms. These unfortunately often aren’t taught brilliantly and hopefully a quick explainer will be of some use to you.

(Adapted from the previous version whose formatting TSR decided to keep screwing up lol)