Question

what is meant by constant independent of initial velocity?

Answer 1

It means that the value of that constant is not going to change with the value of initial velocity.
For eg if the value of that constant is 10 when initial velocity is 10m/s, then even if the value of initial velocity change to 5 or 6 or 1000 or whatever value, the value of constant will be 10 itself.
In a projectile, the time of fall is independent of initial velocity