Hospitals announce that there are not enough nurses available to keep them fully staffed. economically speaking, what does this announcement mean?

this announcement means that: the market wage for nurses will eventually rise to the market-clearing wage

In economy, Market-clearing wage refers to the level of wages/salary which average workers view as fit for certain type of jobs.

When the jobs available in the market do not fulfill the market-clearing wage, the employers wouldn't be able to attract enough people that are willing to spend their time on the job.
This will hinder the production on that industry,and  eventually led to the employers raising up the salary for the job in order to attract potential workers.