How Do NASA and other organisations make sure they are at the right spot to pick them up and that the astronauts don’t land in some war zone or remote hard to reach area?
Answer: The earth is 70% water. NASA sends most it rockets from Florida because most of the debris will land in the Atlantic Ocean when leaving orbit. And on the way back they land in the pacific because it’s the largest ocean. They literally just aim for (as Neil degrasse Tyson puts it) “the big toilet bowl of the world.”
Well they know the speed on the planets and the space ship on each axis and they do a sorta complicated arithmetic using mechanical physics and the trajectory of projectiles to calculate when each body will be at a point where they can come in contact. Basically a lot of math and science brought about by their knowledge of the relativity of the planets and speed of everything so just taking the distance and the speed and finding the time and place.
When you fall straight down, you land right below where you started.
When you fall with some sideways movement you will land further away, depending on where you began to fall and in which direction you were moving and your speed.
After doing lots of science over the centuries, people figured out how to calculate those values.
Orbital trajectories and the effects of air resistance are well known and very predictable.
Assuming that you are not in a spacecraft with wings, there is a predetermined point where you fire thrusters in the opposite direction to your motion to slow you down somewhat. That means your orbit starts to decay and intersects with the top of the atmosphere sometime after. That slows you down even more, generating a lot of heat as you compress the air ahead of you. As the atmosphere becomes thicker lower down you lo*o*se your remaining forward velocity and descend the last bit vertically under a parachute.
All those effects are known and can be factored into the calculation as to when, how long and how hard to do the initial reverse thrust burn to get fairly close to your desired landing spot.
They use math. If you are entering the atmosphere at a given time at a given speed at a given angle, then they can predict (within a margin of error) where you are going to land.