As stated, those are Phoenix connectors and they should NOT have been left on the wire when the old people moved out and took their stuff with them. They are hard to buy anywhere and they are in trouble when they go to reuse their gear.
For you, just unscrew the four screws on top with a #1 flat head screwdriver and you will be left with normal bare wire. Then, depending on the new pieces you get for your setup, you will put the wires into that product as normal.
Many pieces of pro-level audio gear have Phoenix type connectors because they terminate clean and have a solid connection point with almost zero chance of wires shorting out. Unfortunately, a sharp tug on the back pretty easily pulls the entire connector out of the piece of gear which can prove to be a huge headache in residential work where you often don't have good rear access to the equipment in use.
You will most likely end up with something that has a 5-way binding post or a spring terminal on it for your wires. If the wires are not labelled with where they are going to, then it is worthwhile to spend some time figuring out where all your speakers are connected to and labelling all the cables permanently so you know where they go to in the future.
