To the best of my knowledge, the initial motivation for defining the concurrence involves the fact that for pure bipartite entangled states, the reduced density matrices of each subsystem are not pure. For example, consider the Bell state
$$|\Phi\rangle_{AB}=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle).$$
It satisfies $\rho_A=\text{Tr}_A(|\Phi\rangle_{AB})=\mathbb{I}_A/2$, where $\mathbb{I}_A$ is the identity operator on $\mathcal{H}_A$. The reduced state $\rho_A$ is not pure and it purity is $\text{Tr}(\rho_A^2)=1/2$. This leads to the definition of concurrence for a pure state $|\psi\rangle$ as
$$C(\psi)=\sqrt{2[1-\text{Tr}(\rho_A^2)]}.$$
For a mixed state $\rho$, one defines the concurrence as
$$C(\rho)=\min_{|\psi_i\rangle}\sum_ip_iC(\psi_i)$$
where the minimization is over all possible decompositions $\rho=\sum_ip_i|\psi_i\rangle\langle\psi_i|$ of $\rho$. The closed form of the concurrence for two qubits
$$C(\rho)=\max\{0,\lambda_1-\lambda_2-\lambda_3-\lambda_4\}$$
is just the solution to the minimization problem.
