The word staff is what is called a collective noun, which is a noun that denotes a single collection of countable objects. As such, it may be treated as either singular or plural (uncountable or countable), depending on context.
You treat it as singular/uncountable when the entire group of included individuals are doing the same thing:
The staff meets every week and reports on their status.
You can treat it as a plural/countable when different individuals within the group are doing different things (even though the noun itself doesn't take a plural form):
The staff specialize in a variety of disciplines and are assigned to different tasks.
(You'll also find regional differences - British English is much more likely to treat collective nouns as plural than American English is.)
In this latter case, you could imagine that this is using "staff" as a synonym for "staff members" and there are a few (but far from all!) collective nouns where the collective noun can be used idiomatically to mean a member of the collection. The ones I can think of, for whatever reason, generally refer to groups of human workers, so maybe this grew out of what is sometimes pejoratively called "management speak."
So you see constructions like this:
There are only a few overnight staff on duty.
All personnel evaluate each other's job performance.
Half a dozen crew are required to staff each shift.
Again, this last usage is not universal for all collective nouns. Most collective nouns do the singular/plural thing, but do not accept this last idiomatic use as a synonym for "member of the collection."
For example, if you're talking about a flock of birds:
The flock lands on the lake's surface. (correct use of singular/uncountable showing a collective action)
The flock scatter in every direction when a predator appears. (correct use of plural/countable, emphasizing that each bird goes in a different direction)
A few flock stay behind and one is caught and eaten. (incorrect - sounds very unidiomatic)