How do you create projective plane out of a finite field? I have heard and read unclear mentions of links between projective planes and finite fields.
Is it possible to construct a projective plane (or a Steiner system) starting out with a field? Could you, for example, construct the Fano plane with help of a finite field?
 A: Does this help?
Let F be a field. We consider the three dimensional vector space over F. Let subspaces of dimension 1 be points. Let subspaces of dimension 2 be lines. The incidence relation is subset inclusion (i.e. if a point is a subset of a line, then you say the point is on the line.)
These definitions satisfy the axioms of a projective plane. You should be able to recover the Fano plane by applying this procedure with the field with two elements.
A: The Fano plane with 7 points is constructed starting with the field consisting of 2 elements.  See Fano plane.
What you seem to be asking for is an *explanation" of the intuition behind the construction of the projective plane.  Here it would be helpful to start with a different construction using "points at infinity".  This was in fact historically the first construction of the projective plane (the construction using homogeneous coordinates in $\mathbb{R}^3$ came later).  
Briefly, what you want to do is start with the usual plane, and add "points at infinity" to it.  There is going to be a separate point at infinity for each "direction" in the plane.  To define the notion of "direction" rigorously, the usual approach is to take a pencil (i.e., a set) of parallel lines.  Each such pencil defines a "direction".  The part about "adding" points at infinity means that there is going to be an extra point on each line (namely, the one corresponding to the "direction" of that line).  The details can be found for example in the book on projective geometry by Hartshorne.
