Skimming the paper (which honestly I am still trying to understand), the paper authors suggest that the "depth of competition" parameter can be understood as "a measure of the imbalance in strength or skill between the average pair of players".

In other words, it's statement about the skill/strength/power distribution of your population of players.

As I understood it, the general model they are working with (Bradley-Terry) is about how to take a set of win/loss measurements amongst a fixed set of participants, and then try to predict the win/loss probability of unobserved competitions (and I guess also allow you to generate rankings).

Intuitively, I think you can imagine how you could take that type of observation and build out a ranking of participants basically through a series of pairwise comparisons on win/loss rates. But then when it comes time to predict the win/loss ratio of an unobserved competition, you're forced to make assumptions about how to model/extrapolate win/loss ratios. This depth of competition parameter would be part of that extrapolation.

EDIT: Just a thought, I think it's fun to consider the language around "depth of competition". The paper uses "deep" to mean there is a strong gradient in strength. But when we often talk about "depth" in sports, we typically say that "deep" teams or "deep" leagues have a swallow gradient. Think about teams having "deep benches".