how exactly did calculus change our understanding of the world?

Before Newton and calculus, the study of the motions of the planets was at best limited to kinematics: you could observe the motions and try to describe them. As to what caused those motions, there could only be speculation. Newton produced a dynamic theory: some simple physical laws that explained the motions as the result of the same forces that act on an apple falling from a tree. Moreover, with the mathematical techniques of calculus and differential equations (which is basically an extension of the calculus), the resulting motions could be calculated and predicted.

From that time on, most of physical science and engineering has been understood in terms of differential equations. The laws of physics give you differential equations that describe the rates of change of various quantities in a system you are interested in. By solving those differential equations, you can predict how the system changes as time goes on.

Newton's laws, discovered more than 300 years ago, describe the motion of all objects that are not too fast, small, or heavy. The discovery of Newton's laws signalled the beginning of physics as a truly predictive science.

Newton's second law is a second order differential equation${}^\dagger$, \begin{equation*} \frac{d^2{\bf x}}{dt^2} = \frac{{\bf F}}{m}.\tag{1} \end{equation*} In words, the second derivative of the position of an object with respect to time, the acceleration, is directly proportional to the net force impressed upon the object, and inversely proportional to the mass of the object. Without calculus and the machinery of differential equations, (1) is one of nature's incomprehensible secrets.

Given the object's initial position and velocity, Newton's laws can be used to predict its subsequent motion. If the initial conditions and the forces are known with great certainty this prediction will describe everything about the motion of the object far into the future.

Every day countless physicists and engineers rely on Newton's laws to find out where the artillery round will land, how will the car crumple on impact, whether the bridge will fall down in a strong wind, and so on.

_{${}^\dagger$Differential equations relate derivatives of a function to the function itself.}

I think you should first appreciate the fact that integrals and derivatives are an integral part of any understanding of life, physics, and even economics.

Start by trying to describe some property of nature or of your everyday life, say, the amount of money you'd have if you invested the money somewhere.

• You might want to know if there was a certain pattern that would enable you to predict how much money you would have in $x$ years from now - hence the study of functions.
• You may want to check your rate of returns - hence the study of derivatives.
• Finally, you may want to calculate the predicted wealth based only on the rate of return - hence the study of integrals.

This of course is but one of the infinite reasons one would need calculus, the links in the comments are a good start for getting a glimpse of these.