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This would be a good question for the Robotics Stack Exchange. Briefly, the function f is a vector-valued function that takes as its input the entire state vector x_t-1, and outputs a state vector x_t. In this case, even though it's a single function in the paper, we can really express it as a set of functions like this:

X = X + X_velocity * time_delta + X_acceleration * 0.5 * time_delta^2

Y = Y + Y_velocity * time_delta + Y_acceleration * 0.5 * time_delta^2

...etc. The only difference from what I have written here is that we have to rotate both the velocities and accelerations by the robot's orientation, since those quantities are given in the robot's body frame.