Projectively equivalent connections We can define projective structure on a manifold in two ways.
First we can define it as a maximal atlas of charts from the open subsets of manifold to the projective space, such that transitions maps are locally the elements of the projective general linear group.
Second, we can define it as a torsion-free projectively flat connection.
Projectively flat connection is a connection which is projectively equivalent with a flat connection around each point of the manifold.
Also, two connections are projectively equivalent when there is a closed one-form such that we can write:
$D'(X,Y) = D(X,Y) + F(X)Y+F(Y)X$,
$D$ and $D'$ are two connections, $X$ and $Y$ are two vector fields and $F$ is our closed one-form.
Why we need to define projectively equivalent connections?
What is it's interpretation and the relation to the projective space and connections?
 A: I assume you are asking why two definitions of the projective structures, one  given in terms of  atlas, and another given as the  existence of projectively flat connection, coincide. 
If two connections are projectively equivalent, then their geodesics (considered as unparameterized curves) coincide, and vice versa. If a connection is projectively equivalent (locally, in a  neighborhood of each point) with a flat connection, then therefore in a certain  coordindate coordinate chart  its geodesics are straight lines. We make an atlas from such coordinate charts.  Since a local mapping that sends straight lines to straight lines is a restriction of a projective transformation, we have the existence of  an 
 atlas of charts from the open subsets of manifold to the projective space , such that transitions maps are locally the elements of the projective general linear group. 
Now, if there exists such an atlas, then there exists a projectively flat connection. One of the way to see it is as follows. Take a volume form $\Omega$ 
on the manifold (assumed oriented but the proof can be generalised for nonorineted manifolds).  Now, it is an easy exercise to see that for any connection $D$ there exists the unique projectively equivalent connection $D'(X,Y) = D(X,Y) + F(X)Y+F(Y)X$ such that the volume form is parallel w.r.t. to this connection. In the charts from your atlas consider the projectively flat connection such that $\Omega$ is parallel. Because of uniqueness, it does not depend on the coordinate chart and is therefore a globally defined projectively flat connection. 
P.S. You definition of projective equivalence  of two connections is nonstandard: one does not require usually that the 1-form $F$ is closed. It does not affect the proof above though. 
