> Time translation symmetry gives conservation of energy; space translation symmetry gives conservation of momentum; rotation symmetry gives conservation of angular momentum, etc
Something about how if you see that processes can be step-wise played forwards or backwards, it implies the certain quantity is conserved? How?
That being said, suppose you have a theory which describes physical processes. Given that this theory must hold true for every variation of some parameters (rotation, translation, time, ...) then you can derive some things which are constant (angular momentum, linear momentum, energy, ...). Now one can spend many hours thinking about the relation between energy and time with fuzzy philosophical definitions or one can just accept that the math turns out that way (given some assumptions of that theory).
The statement that it does not matter whether I do my (thought) experiment today or tomorrow, can be used mathematically deduce the conservation of some quantity which we can intuit is the mechanical energy.
More technically it's a differentiable symmetry: we can do the experiment a day, a week, a year in the future - this time can be varied smoothly.
If you know calculus read the first few chapters of Landau and Lifshitz book on classical mechanics.