Conservation is all about counting. If we know how to count how much energy, linear momentum,, and angular momentum there is in a system, we should always get the same amount for each. There are formulas which have been discovered through experimentation that quantify each of these charactersitics. Using these formulas which describe what is going on, we observe that energy, linear momentum and angular momentum do indeed stay constant in an isolated system. The following is an analogy.
Imagine the system as a giant office building where the people who work there represent energy, linear momentum or angular momentum.
In this skyscraper, there is only one entrance and one exit to the building, and the security guard is always there to count how many people go in and out of the building each day.
Every morning, the security guard counts how many people enter the building, and at night, he again counts how many people leave. As long as he counts the same number of people in the evening as he does in the morning, everything is safe in the office building, and the security guard is happy and can go home.