Why is this the case?
Since Vacancy rate is defined as following,
let $A,Q,U$ denote number of vacancies in the economy, labor force, unemployed respectively.
$$\frac{A}{A+Q-U}$$
Here we can see that if unemployed increase vacancy rate would go up?
Why is there a negatively correlation then? Take the beveridge curve as an example :
Adopting your notation, the vacancy rate at any given time is defined as $A/Q$. There is no mechanical relationship between the unemployment rate $U/Q$ and vacancy rate (A/Q).
They are negatively related because when vacancy rate is high, it means there are many unfilled job openings, and they are unfilled probably because most people already have a job, thus lower unemployment rate. And vice versa.
The number of vacancies does not change with U. It is the proportion that will change. In other words A will not change because U increases, but the ratio will go up.
