How do scientists know the min limit of temperature is -273 degree celsius?

Historically the value of -273.15 C° was extrapolated from the state equation of ideal gas. In fact, theoretically (without considering quantum mechanics) at 0K° gasses would have 0 volume:

$ pV = nRT $

Another way to calculate it is that at -273.15 (or 0K) atoms and molecules have a kinetic energy of 0, in other words they stop any movement, rotation or vibration, and since there is no way to go lower than "staying still" we can say that this is the lower limit of temperature. (In reality this is much more complicated and this is just a very simplified way to see this) Nowadays the value of -273.15 can also be retrieved considering many physical processes.

The value of 0K is demonstrated to be the lower limit of temperature and it is also demonstrated that it's impossible to reach this value. Some statistical mechanics considerations allow for "negative-temperatures" but just for small period of times, with respect to Heisenberg's uncertainty principle.

On the other hand, from a physicist point of view, you question is "wrong" since for a physicist 0K is just 0k and the "correct" question would be "why we define 0C° to be 273.15K°?". In fact the Kelvin temperature scale is based only on theoretical considerations and from "nature behaviour". The Celsius scale instead is just a convention that we use to represent the temperature.

Obviously the "width" of a Kelvin degree is also a convention, in fact in most of the application and equations you would always see the ratio of two temperatures (which is the same no matter the width of the degree you use) or you just the conversion of the temperature to energy through the Boltzmann's constant $k_b$.

The temperature of a macroscopic substance is a measure of microscopic kinetic energy of the molecules. If a sample is cooled until the molecules stop "zooming around", the temperature can be considered absolute zero. Temperature scales where the zero point of the scale corresponds to absolute zero are known as absolute temperature scales, such a Kelvin and Rankine. The Celsius and Fahrenheit temperature scales do not have their zero points set at absolute zero. For Celsius, the zero point of the scale corresponds the freezing/melting point of water (at standard pressure), and absolute zero occurs at a Celsius temperature around -273.15°. Well, theoretically. Absolute zero cannot actually be attained by any system. Even if we ignore the limitations of quantum physics (zero point energy), the inability to reach absolute zero temperature is still true in classical thermodynamics. However, it seems we can cool a system arbitrary close to absolute zero.