Although similar to the budget constraint in consumer theory, the use of the isocost line pertains to cost-minimization in production, as opposed to utility-maximization.
Each straight line segment is an isocost curve showing various amounts of labor and capital whose combined usage would cost a given amount unique to that isocost curve.
The points on an expansion path occur where the firm's isocost curves, each showing fixed total input cost, and its isoquants, each showing a particular level of output, are tangent.