The primary way of “thinking like an economist” is thinking on the margin. What this means is analyzing how things change with the addition of one extra unit (One extra machine increases productivity by how much and costs by how much, for example). This method of analysis helps us to learn about how resources are allocated and help individuals maximize their profits (or, as economists define it, where marginal revenue equals marginal costs).
This is unlike many sciences. In some, if a variable is small enough, it can be ignored. The good economist does not do this. The good economist thinks on the margin and takes that small variable into account.
This is why I get upset when people claim the costs of minimum wage are small (or negligible), so the wage increase should be enacted. Such an argument is uneconomic. Overall, the effect may be small, but for those trapped on the margin, those whom the wage increase does have a cost, for them the cost is not small.
If one only focuses on the big picture and ignores the margins, one runs the risks of misallocating resources.