If those connections are correct, either a stereo receiver with 'A/B' speaker options would suffice, or something newer with all the 5.1 (and better) trappings. You could pick up something cheap just to test- I often use Amazon as such since they have a very liberal return policy (e.g. grab a $30-50 stereo receiver). Or, find the permanent receiver you want to keep and test that, but only via the front (right/left) speaker connections, configuring/assigning the output to 'stereo' only and not one of the surround modes. Keep in mind that 'Zone 2' options such as second audio source playing through patio speakers are good topics to review now so you don't have buyer's remorse. YMMV.
Ideally, you would snap a quick photo and upload so we can review. I often use imgur.com.
I recently moved into a house. On the patio I noticed outdoor speakers. I'm the family room near the floor there is an outlet for banana clips. I believe that goes to the speakers. No where is there a volume knob. I would like to see if these speakers work. What should I do next? Purchase a receiver or amplifier? I would like to play music through my phone as well. Please help!!

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic