Factor models are a well established technique from multivariate statistics, applied in credit risk models, for identifying underlying drivers of correlated defaults and for reducing the computational effort regarding the calculation of correlated losses.
Assume we have two firms A and B which are positively correlated. For example, let A be DaimlerChrysler and B stand for BMW. Then, it is quite natural to explain the positive correlation between A and B by the correlation of A and B with an underlying factor;
We could think of the automotive industry as an underlying factor having significant impact on the economic future of the companies A and B. Of course there are probably some more underlying factors driving the riskiness of A and B

We additionally wish underlying factors to be interpretable in order to identify the reasons why two companies experience a down- or upturn at about the same time. For example, assume that the automotive industry gets under pressure. Then we can expect that companies A and B also get under pressure, because their fortune is related to the automotive industry. The part of the volatility of a company’s financial success (e.g., incorporated by its asset value process) related to systematic factors like industries or countries is called the systematic risk of the firm. The part of the firm’s asset volatility that can not be explained by systematic influences is called the specific or idiosyncratic risk of the firm.
For limited liability companies, default is expected to occur if the asset value (i.e. the value of the firm) is not sufficient to cover the firm’s liabilities. Why should this be so? From the identity

Asset value=Value of equity+Value of liabilities).

and the rule that equity holders receive the residual value of the firm, it follows that the value of equity is negative if the asset value is smaller than the value of liabilities.
If you have something with negative value, and you can give it away at no cost, you are more than willing to do so.This is what equity holders are expected to do. They exercise the walk-away option that they have because of limited liability and leave the firm to the creditors. As the asset value is smaller than the value of liabilities, creditors’ claims are not fully covered, meaning that the firm is in default. The walk-away option can be priced with standard approaches from option pricing theory.
This is why structural models are also called option-theoretic or contingent-claim models. Another common name is Merton models because it was Robert C. Merton (1974) who first applied option theory to the problem of valuing a firm’s liabilities in the presence of default and limited liability.