In classical groups, people project all those linear groups and make it as simple. Why do they so, Why can't linear groups can be restricted to simple instead of projecting on the scalar space.
What does that projection geometrically imply in that PSL, PSpn,(symplectic) , unitary, orthogona; etc
Projection removes central subgroups. "S" removes abelian quotients. Iwasawa's theorem says that: If a group (a) has no abelian quotients, (b) acts primitively, and (c) is generated by the conjugates of a solvable subgroup, then it is simple. Part (b) typically requires the projection.
There are lots of cases where a matrix group does not need to worry about "P" or "S": for example $\operatorname{GL}(n,2)$ is a finite simple group for all $n \geq 3$ and $\operatorname{SL}(n,3)$ is a finite simple group for all odd $n \geq 3$. However, in order to ensure simplicity, one takes "P" and "S", since the abelian quotients and central subgroups exist for many fields and many types of groups.