"The fermentation broth (600 mL) of each endophyte was filtered, and the filtrate was extracted three times with ethyl acetate at room temperature and further concentrated in vacuo to give dried residues prior to antimicrobial assays"
This procedure is referring to chemical extraction, which is a separation process used in chemistry (and biochemistry) to isolate compounds of interest by using two immiscible phases in which the compound of interest has differential solubility.
For example, in a liquid-liquid extraction, often one of the phases contains a polar solvent like water, and the other phase contains a nonpolar solvent like oil. Water and oil won't mix, but after stirring them together and waiting some time, the more polar molecules in solution will tend to concentrate in the water, and the less polar molecules will tend to concentrate in the oil. Because the phases are immiscible, it is relatively straightforward to separate them once equilibrium has been reached.
Ethyl acetate is a fairly non-polar molecule often used in organic extractions. In the process that you described, the result will be that non-polar contents of the fermentation broth will be extracted into the ethyl acetate, and hydrophilic contents will remain in the aqueous phase.
Because this process is never 100% efficient (the efficiency will depend on the concentrations of the solutes and relative affinity for the two phases), it is often repeated several times to improve the extraction.