I think they might be asking about the Android Logcat logs rather than Appium server logs.
The WebDriver API contains a method to retrieve the logs from the test. If you're using Java, take a look at the Selenium Javadoc for this exact method. You'll have to experiment a bit to see what you need to do to retrieve the logcat logs.
Alternatively, if your tests and Appium are running on the same machine, then your device should also be connected with the machine your tests are running on. You can spawn a process using the command adb logcat, and you can read the standard out from this process to read the device's logcat logs.