Being as it was the Sabbath, when menial housetasks such as laundry were forbidden, that answer didn't sit well with me. So I did a bit of research, and the short answer seems to be that since the Babylonian Captivity, when the Jews were separated from their house of worship, the Temple, the Jews had developed a custom of meeting by rivers whenever a formal meeting place was unavailable. I've been unable to find much support for this idea, but it "feels" right.
One thing my research highlighted is that it was required for ten Jewish men to be in a town before a synagogue could be built. Presumably that also meant that in order to continue synagogue meeting, ten men must be present.
Two chapters after the story referenced above, in Acts 18, we learn that Claudius had expelled all Jews from Rome. I think it's reasonable to assume that may have included Roman colonies, of which Philippi was one (Acts 16:12).
So it may be that there were not ten Jewish men available in Philippi to conduct a synagogue meeting, so the few Jewish stragglers in town, or at least worshippers of God such as Lydia (16:14), who as a businesswoman with her main home in another town was able to remain in Philippi, may have met, as the custom seems to have been, near the river.
=== UPDATE ===
According to http://www.zianet.com/maxey/Inter6.htm:
Therefore, in order to help hold the people together religiously, synagogues were established in locations where ten or more faithful men could be found. Where fewer than ten men could be found, a Proseuche ("place of prayer") was set up, usually by a river and outside the walls of the city....and according to The New Testament Greek Lexicon, the word "proseuche" means:
- prayer addressed to God
- a place set apart or suited for the offering of prayer
- a synagogue
- a place in the open air where the Jews were wont to pray, outside the cities, where they had no synagogue
- such places were situated upon the bank of a stream or the shore of a sea, where there was a supply of water for washing the hands before prayer