What does it mean for a distribution to be "consistent with a two rate-limiting stochastic steps"?
The rate limiting step of a reaction is the slowest component of the reaction. So if there are two steps in a reaction, each of these steps will have a rate, and the slowest of those two rates will determine the overall rate of the whole reaction.

As a simple example, consider the equation

$A + B + C\rightarrow AB+C\rightarrow ABC$

If the rate of $A+B \rightarrow AB$ is fast, but the rate of $AB+C \rightarrow ABC$ is slow, then the overall reaction will still be slow.

The figure in your question is showing the distribution of the burst durations, and highlighting the fact that the calcium level has no effect on the burst duration. The second quote is a bit confusing, this is my interpretation:

The functional form which they fitted to the distribution indicates that the expected burst duration is about 70 seconds. I'm not completely sure which stochastic steps they are referring to, maybe translation and subsequent localisation (I think transcription would happen on a different timescale and so could be ignored in this case).

By the way, this paper is a really excellent example of an organism harnessing stochasticity to its advantage in order to synchronise its response to a stimulus (calcium in this case). I just think that it is so cool that the mean response to the signal level can be different to the response to the mean signal level - so counter-intuitive to me. And so much fun!!