Why GDP is converted to 2010 $US constant in international studies?
They do that in order to express GDP in real terms. This is incredibly important because nominal GDP is very deceptive since money constantly changes its value. US dollar in 2005 is completely different dollar from US dollar in 2021 so GDP expressed in 2021 US dollars can't be compared to GDP expressed in 2005 dollars. It would be like trying to compare distances in imperial and metric units without any conversion.
For example, suppose that nominal GDP in year 2020 was 200 and in 2021 nominal GDP was 400. Does that mean that economy now produces twice as many goods and services? Well we do not know. It is entirely possible that the economy produces more or it might produce same amount of goods and services or less. This is because nominal GDP is combination of real GDP and price deflator:
$$\text{Nominal GDP} = \text{Real GDP} × \text{GDP Deflator}$$
Problem is that if you look at nominal GDP you never know if it increases just because prices (i.e. deflator) changed or because we actually have higher output real output, and consequently nominal GDP is next to useless for comparison of output across time.
However, while nominal GDP is useless we can still make comparison across time by calculating real GDP where we get rid of the price level by dividing the nominal GDP with the deflator.
$$ \frac{\text{Nominal GDP}}{\text{GDP Deflator}} = \text{Real GDP} $$
Real GDP can serve as a basis for comparisons across time because real GDP accounts for the change in prices by being expressed in constant dollars (dictated by the year for which deflator is set 1). You can have a look at Mankiw Macroeconomics pp 25 to learn more.
why the chosen year is 2010 in this case?
You always have to choose some year to be a base year for deflator (year when deflator =1). You can choose any year you want. In this case they choose 2010. There might be some situation where you want to set deflator to some specific year for sake of making informative graphical illustration, but other than that the choice of base year is completely arbitrary. If you run empirical regressions with GDP being expressed in 2010 dollars you will get the same results as when the GDP is expressed always in 2005 dollars.
