The output from the probe is swallowed by the Kubelet component on the node, which is responsible for running the probes (source code, if you’re interested). If a probe fails, its output will be recorded as an event associated with the pod, which should be accessible through the API.
The output of successful probes isn’t recorded anywhere unless your Kubelet has a log level of at least –v=4, in which case it’ll be in the Kubelet’s logs.
Feel free to file a feature request in a Github issue if you have ideas of what you’d like to be done with the output 🙂