For a Bayesian a random quantity is just an unknown one. For example a coin not yet flipped is random (because I don't know which way it will land), and so is the population of Colorado (because I don't know what it is). Frequentists treat randomness as an inherent property of things, so that the coin flip would still be random (because it's not predetermined) but the population of Colorado isn't (because it's already fixed).

So given the problem of estimating the population of Colorado, a Bayesian would just hand you back a probability distribution (i.e. tell you how probable each population was). This option wouldn't be available to the Frequentist, who would refuse to put a probability distribution on a variable that wasn't random. Instead the Frequentist would give you an estimate and then tell you that the algorithm that generated the estimate had desirable properties, like being "unbiased".