With one plane you divide space into two.
With a second plane you divide those two subspaces into four. The intersection of the two planes is a line.
With a third plane you can divide all four subspaces into eight. The intersection of the line and the third plane is a point.
So you have three planes, three lines (one for each pair of planes), and a point.
The fourth plane can intersect the previous three planes. So that's three more lines and four more points.
In fact, every plane intersects every other. This is optimal.
Now three planes cut the fourth into a triangle surrounded by unbounded regions of the fourth plane. All four planes are thus cut into triangles because every plane intersects every other.
Each triangle meets each other at a line segment, and each group of three meets at a point. This is a tetrahedron.
So space is cut into a tetrahedron surrounded by unbounded regions. There is an unbounded region for every face, line segment, and point.
From each face extends a region bounded by the plane that includes that face, and the three other planes, and otherwise reaches to infinity.
From each line segment extends a region bounded by the two planes that include that line segment, and the two other planes, each of which cuts the first two. It otherwise reaches to infinity.
From each point extends a region bounded by the three planes that include that point, and otherwise reaches to infinity. The fourth plane opposes this point just like the base of a pyramid opposes the apex.
So that's a region for each face, line, and point, plus the tetrahedron. That's 4+6+4+1 = 15.