How do we know the order of the new Trappist-1 planets?
Using Kepler's laws.
We can determine the orbital period, $P$, each planet has by just looking at the time between transits for a given planet. Each planet has a distinct transit so its easy to distinguish which planet is transiting and we can calculate a period pretty easily.
Kepler's law then tells us that if you're orbiting with a specific period, you have to be orbiting at a specific distance (based on parameters of the system). The general equation is given by
$$P^2 = \frac{4\pi}{GM_{star}}a^3$$
Where $G$ is just the gravitational constant, $M_{star}$ is of course the mass of the star and fairly well known, and $a$ is the radius at which the planet orbits. Simply calculate $a$ for each of your planets and put them in the correct order. You could even do this yourself using data from the paper.