Why Does Resistance of a Conductor Increase With Length?

If I take cross section close to beggining of the conductor, charges which start moving on one end don't experience as many collisions when they get to that cross section close to the beggining as they will when they come to the other end of a conductor. It seems that resistance should increase from one towards the other end of an conductor.

You seem to have answered your own question. You describe considering a shorter length, and correctly come to the conclusion that there would be less resistance.

What does length of a conductor have to do with how many charges will pass through any cross section of a conductor in unit time?

It doesn't. You are right; for a constant current, a longer resistor will have the same number of charges flowing through it per unit time (as is evident from the definition of current). However, you will need more power (larger potential difference) to keep that current constant as the resistor gets longer. So there is more resistance in a longer resistor.

Most common explanation is as wire is longer electrons experience more collisions as they move from one to the other end of the wire.

Yes, and because of that it takes more work per unit charge (more voltage $V$) to move the same amount of charge per unit time (same current $I$) from one end of the wire to the other end of the wire the longer that wire is. Then, from Ohm's law

$$I=frac{V}{R}$$

It follows that if the voltage $V$ must be greater to achieve the same current $I$ in the longer wire, the resistance $R$ of the longer wire must be higher.

Hope this helps.

Lets make it simple to visualize. Consider a pipe and your task is to flow water through it. You apply some sort of energy to get water from one side to travel to the other end. This energy is potential energy difference or what we call as Volts. If the pipe is longer, for same water amount, you need higher energy, that means resistance offered by the pipe as a whole to water flow is more. Hence more resistance. If the pipe length is small, then less energy is required to transport same amount of water. Hence less resistance. Here the amount of water that flows per unit time is analogous to current.