With increase in cross sectional area the number of atoms with which electrons will collide will also increase in the same proportion. So basically the resistance shouldn't change with increasing wire thickness. Then why is it inversely proportional to cross sectional area?
The question points out to the distinction between the resistance and the resistivity. Resistivity characterizes the material and remains constant whatever is the cross-sectional area. Resistance characterizes the total flux of the electrons, which is increasing proportionally to the number of holes between the atoms where the electrons can pass (adopting the simple model of resistance implied in the question).
One can think of it as water flowing on a rugged surface: ifthe density of the obstacles is the same, the density of the flux will be also constant, but the total flux is the bigger, the wider is the flow.
In a rough physical sense, you can define resistance as the function of number of collisions electrons undergo during it's motion through three conductor. When the cross sectional area increases, the street for the electron gets less cluttered up and hence it can move with lesser number of collisions. However, that is a very naive explanation. A more general explanation would be to say that by definition, we can write :
And hence the inverse proportionality relation is responsible for increase in area, decrease in resistance property.
there is an intuitive answer that requires almost no math. We know that electrons can travel through a wire. For a given wire of given diameter (and given length), we can measure the current or amount of electrons passing through the wire per unit time. There is a force driving this movement of electrons that we call a voltage difference between where the electrons enter the wire and where they exit the wire.
Resistance is simply the ratio of the magnitude of the voltage difference compared to the amount of electrons that exit the wire per unit time. Regardless of any of the physics that determine this ratio (resistance),it is obvious that two identical wires (same material, length and diameter) with the same voltage difference from one end compared to the other, will transmit the same amount of electrons per unit time. Obviously, two such identical wires have exactly the same diameter or cross-sectional area. In this case, two such wires will transmit twice as many electrons per unit time compared to only one wire.
Assuming no other physical phenomena related to increased cross-sectional area affect the transmission of electrons through the wire, then a single wire of twice the diameter or cross-sectional area of two otherwise identical wires, should transmit twice as many electrons per unit time. In other words, the wire resistance should decrease directly proportional to the diameter or cross-sectional area of the wire.