Follow the five steps in the question. These steps lead you to the required proof.

(1) Here you are glueing 2 polygons along a common edge and the area of the 'sum' of the polygons is the sum of the areas of the individual polygons. Decide which lattice points on the common edges become interior lattice points to the new polygon and which are on the edge of the new polygon and make sure none of these is counted twice.

(2) Proving Pick's Theorem for a rectangle simply involves counting lattice points.

(3) Now deduce Pick's Theorem for right-angled triangles.

(4) Now you know Pick's Theorem holds for rectangles and right-angled triangles deduce that it holds for the general triangle.

(5) Finally deduce Pick's Theorem for the general plane polygon using the earlier parts of the proof.