You're right, a vector can't be parallel to the planes - so what you actually are finding is a vector in the same direction as the intersection of the planes.
I'm sure you learned about cross products, which is what you have to use to find this line. That matrix, when you take the determinant of it, is the cross product between the two planes' normal vectors.
A normal vector is the vector that is perpendicular to the plane. Taking a cross product between two vectors gives you a vector perpendicular to those two vectors. Sooooo when you take the cross product between the two planes' normal vectors, you get a vector that is perpendicular to both those normal vectors. So that means this vector you get after computing the determinant of the matrix will be "parallel" to both planes.
The "i" "j" and "k" are normal vectors of length 1 in the x, y, and z direction. If you have issues actually taking the cross product/determinant of that matrix, this is a good site: Pauls Online Notes : Calculus II - Cross Product