When that options enabled, the plugin will make an AJAX request every ~30 seconds that checks if the background task that processes posts/links is running and starts the task if it isn’t.
The server load depends almost entirely on how many unparsed posts and unchecked links you have at any given time. If it’s just a couple dozen, the load will be negligible even with that option enabled. If it’s thousands, you might notice some performance degradation on weaker servers.
Anecdotal example : My personal blog runs on a low-end VPS and contains ~4000 links. Re-checking them all in one go has no visible performance impact.
My host’s server crashed because a log file got overfilled with POST PAYLOAD related errors orginating from requests by you plugin.
It’s not the server load that caused the problem, but Broken Link Checker was indeed making requests every second or two, which caused the log file to inflate rapidly since the server setup was such that each request generated an error.
Switching to hourly check seems to have solved the problem on my side and my host changed some things on his side too, so I think we got things under control now.