On the wiki page for Grothendieck group, the first sentence says the Grothendieck group is the "best possible" way to construct an abelian group from a commutative monoid.
What actually does this mean formally?
Statements like this usually refer to universality. Note that given a monoid $M$ the Grothendieck group $G$ comes with a map of underlying monoids $s:M\to G$.
Now given $H$ an abelian group and $f:M\rightarrow H$ a map of underlying monoids. Then there exists a unique group homomorphism $g:G\to H$ making the diagram commutative, i.e $g\circ s=f$. This also makes $G$ unique up to isomorphism.
You will find universal properties of that kind at many places in mathematics.
Addendum: The following more or less reflects my personal taste and therefore might or might not be helpful. Whenever I see a universal property I don't fully understand, I convince myself that it makes sense in a simpler setting I can grasp better. So if you want to replace the words monoid by subset of $\mathbb R^n$, homomorphism by inclusion and group by closed set. What is the closed set which approximates a given arbitrary set in the best possibly way? Does this satisfy a similar universal property?