I get the idea that physics wishes to study changes to discover new rules. But why is everything related to rates? Acceleration,Velocity? Could we use something else apart from these? What can you think of istead of velocity(rate) to describe motion? Why rates?
We try to discover laws that help us predict the future location of an object using information about it's current state. This means that we want to know how it's motion changes with time, so we can calculate the next step and the next etc.
Newton's laws describe how the position of an object changes with time, in order to predict it's location at later times. This can be calculated numerically, or by solving differential equations (integration).
If you agree that a lot of physics deals with how things change over time (example how do the planets move over time, or how do the waves of the ocean propagate in time etc), then the answer is calculus. Because VERY VERY roughly speaking, if you want to understand how something changes, you should first try to understand "small changes", but understanding "small changes" is really the subject of differential calculus. Studying "big changes" is then the subject matter of integral calculus (putting together all the small changes to recover the total change). The connection between these two is given by the fundamental theorem of calculus (if you're dealing with the one-dimensional case).