So, I understand that True air speed is the motion of aircraft relative to air around it. And, indicated airspeed is speed arising due to dynamic pressure.
Considering a true airspeed of 100 knots, that means the air molecules are striking the aircraft at 100 kt/ vice versa. Shouldn't this make true and indicated airspeed same?
Also with constant indicated airspeed, why does true airspeed increase with altitude? Shouldn't it remain same?
Air pressure reduces with altitude, so while an individual molecule is striking the the aircraft at 100kt, there are fewer of them. This means there is less pressure placed on whatever surface is being struck.
The difference in indicated speed is simply one of a physical limitation with the air speed indicator - air speed indicators rely on dynamic pressure/ram air to give a readout. Because of that reduction in pressure, the readout begins to diverge from the true, physical speed as air pressure differs from the reference pressure of the gauge.
It would be possible to create an airspeed indicator which adjusts for this, in the same way as the altitude indicator does. However, as it happens, the planes flight characteristics also changes as pressure reduces in much the same way as the air speed indicator does. Because of this, for pilots, the IAS is the important speed outside of navigational purposes.
Hence, all design speeds are written in Indicated Air Speed and can be referred to at any altitude.
Think of it this way. Your airspeed indicator is not directly measuring the distance traveled of any object, aircraft nor air molecules, over time. Your airspeed indicator is a pressure sensitive device that measures ram air pressure (dynamic pressure) versus static pressure. It converts that measurement into a meaningful value of airspeed.
That ram air pressure device is dependent on the density, or more precisely, the mass, of the fluid it measures. The less mass there is to measure, the less ram air pressure measured. The Earth’s atmospheric pressure decreases logarithmically with altitude due to the weight and density of air as a fluid. Half of Earth’s atmospheric mass is below 18000 feet MSL. It’s density decreases rapidly from there. If the same pitot tube at the same velocity were placed in water, it would measure a drastically greater IAS due to greater mass of water in the same volume of space. Even if the aircraft remains at the same actual velocity, the IAS would change with the mass (therefore density) of the air entering the pitot tube. This is because there are fewer air molecules In a given volume of air to create the mass.
A real world analogy is the difference between standing in a brisk wind versus standing in a moving current of water. The moving wind would have to be moving at a very fast speed in order to exert enough pressure or force to overcome your inertia. The moving water of the current does not have to be moving nearly as fast at all to do the same.
Dynamic pressure is
$$ q = ½\varrho v^2 $$
where $q$ is dynamic pressure, $\varrho$ is density and $v$ is velocity (a.k.a. true airspeed). The important bit here is that it is proportional to density and since density and pressure are closely related and pressure decreases with altitude, so does density
For the intuition of particles hitting the surface, note that pressure is force times area and force is mass times acceleration. So lower density (density is mass per volume) means less force and less pressure for the same velocity.
TL-DR: IAS is not about speed but about dynamic pressure. You can read it as "This is how fast I would need to fly at sea level standard conditions in order to get the same dynamic pressure"
As Jan Hudec pointed out already, indicated airspeed is computed from dynamic pressure. That itself is computed from total (i.e. stagnation) and static pressure, and the difference between both.
For (lossless) incompressible flow (i.e. anything slower than half the speed of sound, you might remember Bernoulli's equation:
$\frac{1}{2} \rho_\infty v_\infty^2 + P_\infty = P_T$
What is measured in flight is $P_T$ (total pressure, at the stagnation point) and $P_\infty$ (static pressure of undisturbed flow, measured the sides of a pitot-static tube). The difference between both is called dynamic pressure ($q_\infty$), and allows you to compute velocity:
$v_\infty = \sqrt{ \frac{2q}{\rho_\infty}} $
...except you need to know the density. Density mostly follows the ideal gas law, which means that it varies with pressure and temperature, but it also changes with humidity.
In actual aviation, such effects can be taken into account for the purposes of navigation. You can work out the actual airspeed plus windspeed fairly well -- but for the purpose of designing an aircraft, dynamic pressure is actually more important than true airspeed, and also easier to compute. So when designing an aircraft and calculating loads, as well as as minimum and top speed, indicated airspeed is actually more useful than true airspeed, as it scales directly with dynamic pressure, and all aerodynamic loads on an aircraft scale directly with that. That is why for example take-off, landing and stall speed are quoted as IAS. The important bit is not how fast you're moving but whether you're generating enough dynamic pressure to stay in the air (or take off, and so on...). When you're taking off in Lhasa you need the same IAS as in New York, but you need a higher TAS in Lhasa to compensate for the thinner air.
Now, for compressible flow (very roughly: going faster than half the speed of sound), the incompressible Bernoulli equation is not that accurate anymore. Prandtl-Glauert transformation can kinda make it work a bit further, but really it makes a lot more sense to stop thinking about velocity and use Mach number instead, and the isentropic state change equations. They result in this handy equation to work out velocity:
$v_\infty = M * a = \sqrt{\frac{\gamma-1}{2} \left(\left(\frac{P_T}{P_\infty}\right)^{\frac{\gamma-1}{\gamma}} -1\right)} \sqrt{\gamma R_s T_\infty} $
($\gamma \approx 1.4$ for air, $R_s$ is the specific gas constant, and $T_\infty$ is static temperature)
It's perfectly feasible to use this to work out the true airspeed (given some information on temperature and humidity, and calibration of the probe given that it doesn't quite see undisturbed flow) -- but except for navigation, IAS is still used when designing and flying an aircraft.
Reason: It's all about dynamic pressure. Someone who uses IAS is usually actually interested in dynamic pressure, not how fast the aircraft is moving. Using a velocity rather than a pressure is done out of habit (because for many decades, IAS was the only number available to pilots), and because it actually approaches true airspeed at lower Mach numbers and altitudes (i.e. for take-off and landing), so most pilots already have a "feeling" for those numbers. That means it makes sense to keep using IAS instead of straight-up pressure.
This is why even transonic aircraft have not just take-off, landing and stall speed quoted in IAS, but even top speed is limited not just in terms of Mach number but also IAS. This means: You're fine to fly at Mach 0.85 at 11km altitude but if you need to make an emergency descent you can't stay at M=0.85 all the way down to 2km because the aircraft was not built to deal with the dynamic pressure you'd get in the much thicker air at lower altitudes if you did not slow down.
