How can I capture other computers traffic in Wireshark on a WiFi-network?

Solution 1:

You need a network card that can be set in 'promiscuous mode' which I don't believe the MBA allows.

You may have better luck using ADB to get logs via usb debugging in this specific case but in general you're going to need different network hardware.

Solution 2:

You'd need to capture in promiscuous or monitor mode; not all Wi-Fi adapters and their drivers handle promiscuous mode, but many of them handle monitor mode, including the adapters in various flavors of MacBook and the OS X drivers for them.

If the network is encrypted (WEP/WPA), monitor mode traffic will be encrypted, and you will have to arrange that Wireshark can decrypt it.