Is there a simple way to understand how scientists estimated/calculated the following percentages?
The total energy density of the universe can be inferred from the geometry of space-time, which in turn can be inferred from the observed size of microwave background fluctuations.
The amount of normal matter can be extrapolated from direct observation, the amount of dark matter indirectly via gravitation, the rest is dark energy.
See this site from NASA for a basic introduction.
I am afraid it is not simple. The normal matter would be the visible matter i.e. something directly observable. As for the Dark Matter, various observations point to its existence. One of them being the galaxy rotation curves. This is when the stars rotating in the galaxy are moving a lot faster than they would if there were only the visible galaxy matter creating the gravitational pull. You can also study galaxy clusters which will give you an idea of how much Dark Matter there is. Dark Energy is even more complicated. The evidence for that stems for observation which show the Universe to be expanding. Based on the expansion rate one can deduce how and how much of this Dark Energy is around to cause such an accelerated expansion.
From studying the motions of clusters of Galaxies, you can calculate its motion and therefore estimate its mass.
Then study clusters of clusters of Galaxies and you can estimate the total mass and the rate of expansion.
After that it is simple psychology:
Normal (visible) Matter 4%, Dark (invisible) Matter 75%, Dark (expansion) Energy 21%
Give something a name and number and people will believe anything.