If you look at e.g. 4 min 27s on the video you will see he is using Remote Desktop Connection to an IP address.
You need to set up Remote Desktop Protocol (RDP) on the Pi, then use an RDP client on the computer - not sure if he's using Windows or Mac. Other types of remote desktop software with other protocols are available. This guide discusses first VNC (...
Rpi + ADC MCP3008 read sound from sound sensor get ADC results 0 and 1023
can record the sound and play it through speaker?
Yes, no problem, if you can bear poor HiFi quality. Since Rpi is an educational tool anyway, it should be a very good learning exercise.
Let me make some random comments.
You are using a "sound sensor" ...
If the relay pins are a simple contact closure (which seems likely, but no one can confirm - although it is easy to verify with a multimeter) then you can use any button code.
Connecting to ...