When you use “kubectl proxy” from your server it uses your current kubeconfig credentials to authenticate to the API server and proxies your API server URL to the provided “kubectl proxy” port (default 8001).
Without “kubectl proxy” you would usually -
Do a curl to API server URL with required key/cert info (args - --cacert, --key, --cert) OR
Use kubectl commands (which again uses cert info from your kubeconfig - default location ~/.kube/config)
Now, within a pod, you would need your kubeconfig file and kubectl installed to use kubectl proxy the same way.