Consider a circuit consisting of a variety of linear and/or non-linear elements and independent voltage and/or current sources. Now imagine a dependent current or voltage source is added to an arbitrary part of the circuit, and assume that the voltage or current the source depends on is non-zero. The dependent source would generate a non-zero current or voltage which would affect the rest of the circuit.
It's easy to imagine that the dependent source would affect the current or voltage it depends on, which in turn would change the magnitude of the dependent source, which in turn would affect the current or voltage it depends on again, essentially creating an infinite loop of the dependent source and the voltage/current it depends on increasing each other until both reach infinity.
An analogy might help. A man tells his friend that for everything dollar he has, he will give him another. (Analogous to the dependent source "agreeing" to "contribute" an amount that is a function of what another part of the circuit "has"). His friend starts off with $10, so the man pays him $10. His friend now has $20, so the man now pays him another $20. The man now has $40; this continues on until the man is paying his friend an infinite amount of money.