An Airspeed Indicator is simply a differential pressure gauge with its dial marked off in knots instead of PSI.
It gives an airspeed indication that is determined by the dynamic presssure that it is sensing.
To do this it captures Total Pressure (which is static pressure plus dynamc pressure) then subtracts static pressure from it. In this way it isolates the dynamic pressure inside its capsule.
Any change in dynamic pressure will expand or contract the capsule and it is this movement which determines the position the needle on the dial.
This means that every time it senses a given dynamic pressure it will give the same airspeed indication.
Dynamic pressure = 1/2 Rho Vsquared where Rho is air denisty and V is TAS.
If we ignore the niceties of instrument error and pressure sensing error then Airspeed Indicators are calibrated so that IAS = TAS at ISA mean sea level.
Now lets imagine that we climb at constant indicated airspeed. This means that we will be climbing at constant dynamic pressure.
But as we climb, the air density, Rho decreases, so we need more Vsquared to give the same dynamic pressure.
At 40000 ft for example, Rho is about 1/4 of its sea level value. So Vsquared must be 4 times its sea level value if dynamic pressure is to be constant.
This means that at 40000 ft the TASsquared is 4 times the IAS. Which means that TAS is twice the IAS.
If we try putting different values for Rho into the equation we can see what must happen to V to keep Dynamic Pressure constant.
For example
Dynamic Pressure = 1/2RhoVsquared
let's start with Rho = 1 and V = 20
1/2 x 1 x 20squared = 200.
So we have 200 units of dynamic pressure.
If density reduces to 1/2 then to get the same 200 units of dynamic pressure we need
1/2 x 1/2 x 28.28squared = 200.
If density decreases to 1/4 then to get the same dynamic pressure we need
1/2 x 1/4 x 40squared = 200