Seems like your startup disk is damaged & the OS cannot use it. There are two main pieces to a default installation, a 'recovery partition' & the 'system partition'. The Mac seems to be unable to find either of those so it is falling back to use 'Internet recovery mode' instead.
The internet recovery system is launching, that requires a stable internet connection and a WPA/ WPA 2 wifi network.
OS X: About OS X Recovery - Apple Support
You can also connect an ethernet cable to avoid the issues that wifi can introduce (slow performance & bad reception/ dropped connections). Ethernet is really the best option.
Sorry you my need to take this Mac to an Applestore or contact Apple support. You may lose data if Apple repair it, inform any technicians at a store if this is your only copy of this data they generally do not do data recovery at Apple stores - it is a specialist task.
You can use Apple diagnostics, but it doesn't try to fix anything on the Mac, it just tries to tell you what is wrong…
Using Apple Diagnostics - Apple Support
One final option is to use Target disk mode & another Mac. It may allow you to read some files & recover them, but I doubt it. It seems like your disk is not responding at all.
https://support.apple.com/kb/PH10725?locale=en_US
You need to have a suitable cable and two Macs to use it.