ill explain it how it made the most sense to me while learning it (if you want a more of a physical reasoning i think schutz does well). the book i am referencing is https://www.amazon.com/Special-Theory-Relativity-Mathematics-Students/dp/9810202547 .
let V be a four dimensional vector space with bases {e1} and {e2}, and let c>0 be a real number. define a 'minkowski scalar product' (with respect to a basis {ex}) as (u,v){ex}=-c^2 u^0 v^0 +u^1 v^1 +u^2 v^2 +u^3 v^3 .
call two bases {e1} and {e2} of V equivalent if (u,v){e1}=(u,v){e2}
you can show that if {e1} is a basis of V, and you define a new basis {e1} of V by the lorentz transformation equations, then these bases are equivalent. using the postulates of relativity, you can then show that this c that was mentioned is actually the speed of light, and this inner product is called the spacetime interval.