As you inferred, this is related to the concept of diversification as a risk-mitigation tool.
In short, think of $\rho$ as representing some risk measure, and $\rho(x)$ as the risk of asset $x$ under that measure. If subadditivity holds, then the risk of holding assets 1 and 2 simultaneously must be less than or equal to the sum of their individual risks: $\rho(x_1 + x_2) \leq \rho(x_1) + \rho(x_2)$.
For example, volatility (standard deviation) is a subadditive risk measure. We know this intuitively from diversification: a portfolio is less volatile than the sum of its component volatilities.
As it relates to finance, subadditivity is one of the four axioms characterizing "coherent" measures of risk. This class of risk measures was introduced in Artzner et al, 1998, see the bottom of page 6. Think of these as risk measures with desirable properties that won't be subverted by strange-behaving portfolios. It's important to note that subadditivity is not a statement of fact -- it's easy to define risk measures that are not subadditive -- but rather an axiom that risk measures must satisfy in order to be coherent.
Artzner describes subadditivity nicely as the idea that "a merger does not create extra risk," and lists a number of practical points which follow from it. One interesting one is that if risk were not subadditive, then a person wanting exposure to asset 1 and asset 2 would be better off opening a separate account for each asset, as the (risk-based) margin requirement would be lower than if he held both in the same account. (Note this can be seen as a very literal interpretation of the right hand side of the equation.)
The most (in)famous risk measure that does not satisfy this axiom is VaR. The VaR of a portfolio of two assets can be greater than the sum of their individual VaRs. This is because VaR is a quantile-based measure; see the Artzner paper for examples.