On the top line is given what people like Galileo knew. Its called Gravity. It says as you go away from center of gravity your energy increases as the force of attraction you experience goes on decreasing with the increase in square of r, r is separation of you from the center of the gravity creating object such as earth. So Newton could have differentiated it. (why Galileo isn’t ascribed to derivatives and calculus? he definitely had to understand it to see gravity forces? thats the kind of questions historians better be talking about) Now if I differentiated the top what I get is the 2nd line. (I took derivatives, which are ratio of small changes in different variables) So I get on 2nd line on Left hand a fractional change in energy dE/E (eg did energy change 1% or 5 % ? .. ) On the right hand side you have a fractional change in r, which is a change in gravitational deflection (eg did the object moved 1% or 3 % or 5 % on top of what deflection it had if the energy didn’t change?)
And on total right you have the quantity that lets you decide the answers to these questions. the total right is v*dt/r which says in ordinary situations like ours the change in time dt in which you are seeing what deflection takes place (change in position r) could be compared to that deflection, but the v, the speed at which the object is traveling is so small compared to the speed of light ( a misnomer its actually speed of photon which is different) that the total right hand side is zero. In other words the fractional changes in energy and deflection or length and time and frequency everything is zero for such ordinary objects like cars, satellites, cricket ball etc moving under influence of earth or say sun. But for things like electrons, photons, protons which are so so small that they would essentially move at very high speed compared to the maximum c, the speed of light and no more the right hand sides or left sides are zero.
That means there appears to be relative changes in everything we know when things gain speed. Due to moving in high speed or being placed in different locations in the gravity region of such objects as earth or sun you would experience the rate at which (definition of) your energy changes, your length changes, your frequency changes or your time changes. they are no more absolute values but depend upon your or their speed and their locations in the Universe if those locations are differently gravitational. This is called theory of Relativity.
This relativity could have been recognized to be Newtonian or Galilean in nature actually but in their times no fast moving objects were there , neither was the fact that clocks would time differently if placed on higher vs lower locations of gravity were known. These time effects are called time dilation since a fractional change in time due to motion or location in gravity regions is a fact that the units of time has changed. Since the high gravity (low energy) would mean time slows its actually a time dilation effect where the unit of time has become larger than when the gravity was less intense (hence more energy). In this, the reverse situation that is, its a time contraction (so time runs faster, the debts of energy have been released and one is feeling more relaxed and rejuvenated and runs faster, same with time). But for motion when the speed is getting higher there is time contraction and when speed is getting lower there is time dilation. This is to conserve the total amount of energy or in-fact to conserve the amount of space or time etc that are available. You may not have more space/time available to you in any point of Universe and nature automatically dilates your space/time so that you cover them in longer time or slower pace. (or in longer or smaller strides, that means length contraction is the fact that you contracted your stride simply because in the next instant less space was available).
Also one needs to see space and time as space-time so as to get more and more correct description of any situation and that embodies an actual treatment of Relativity Theory as is known today (or even from Einstein’s time) . These are the 4-vector formulation, in terms of tensors and so on.