A few days go I launched my old phone with old version of Google Play. I launched it, but in few seconds application crashed. When I launched it again, a new version (this one that I have on my current phone) was shown infront of my eyes. I'm asking, how? I have a theory that Google Play checking content of server and when detect a new apk, it install it in the background. Am I reasoning it right?
When you're connected to Wi-Fi and Google Play Store is running, it will automatically download any updates for Play Store and Play Services. If you have something like a network monitor, you'll occasionally find Google Play Store occupying large bandwidth and stops after tens of MB.
Updates for Play Services will be installed immediately while updates for Play Store will be installed after you quit it (by pressing back multiple times or home). As noted by @
Matthew Read, apps with system privileges do not require user intervention while installing or updating apps.
Yes, you're correct. Google Play Services checks for and installs new versions when needed. As acejavelin mentions, Google services are privileged apps and do not need user interaction/permission to update.