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. According to classical economists, wages are determined by a model called the market theory of wage determination, a model similar to that of the supply and demand. More specifically, this model suggests that when workers sell their labour, the price that they charge depends on the number of the available workers (supply) and on the number of the needed workers (demand). In this example, we can conclude that the number of available nurses is not enough to meet the demands of the system. Therefore, the available nurses can sell their labour at a higher price.

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.