We don't really know. It must have been done by aqueducts, and we know which ones fed the area, but the site of the Flavian Amphitheater was completely renovated at and under ground level under Domitian, and as far as I am aware no archaeological evidence of the actual means of how the site was flooded remains. We know that naumachiae were known for rapidly changing land shows and water shows, but no text refers to how it was done. However, the Flavian Amphitheater was in fact rarely used for naumachiae. Most naumachiae were done in natural or artificial basins, which could be flood and drained from the Tiger or sometimes aqueducts (Augustus flooded his from the Aqua Alsietina since it's water was found to be nasty) or in the case of Nero's, in temporary amphitheaters built near the Tiber that could be drained easily. The Flavian Amphitheater was only used for naumachiae a couple of times, and we don't really know how or what it looked like. We assume that the Flavian Amphitheater was flooded with the floor serving as the basin's bottom, but it need not have been so. There's a surviving basin for aquatic shows (probably fish and sea monsters, it's too small for naumachiae) dug into the ground of a smaller amphitheater at Merida--it's below ground level so the idea would have been to put a false floor on top when land shows were staged. Since we have no idea what the Flavian Amphitheater looked like below ground before Domitian added the basements we see now there's no way of knowing--the fact that the naumachiae were only staged early in the amphitheater's life suggests that whatever Domitian did he made it difficult to flood the amphitheater (and certainly no remnants of a drainage system survive)