exactely what would you like to know?
in a very simple schematic is like this:
hardware + blynk lib (any mcu with internet connectivity) <-> blynk (cloud or local) server <-> mobile blynk app
the hw sends and receives data to / from the server, (even while the phone app is not active), and also the phone app can communicate with the server while the hw is offline. whichever part when first goes online will receive the data / command from the server.
note that by default the blynk lib is blocking, so if can not connect to the server in setup function, will never enter the main loop (this can be overriden with manual config)
also note, that if using bluetooth for communication, the above schematic is different:
the hw will no longer need server access, however the phone still needs server access for initial authentication.
(hopefully this will change in the future.)