Change the date in the top/right corner of Analytics for a single day on that spike, then sort by highest views per page, that will at least show which page was bringing in the traffic. If your targeting one main keyword per ranked page, then take that highest viewed page keyword & search for it on Chrome Incognito, are you still ranked? I guess your saying all that traffic/data is from the SERPs.

Well this is normal in search results too

It is so normal of getting fluctuations in a graph & BTW, you opted in showing graph as month order.

Since Google updates it's search quality every three day/week/month, it is common in getting fluctuations, check for backlinks getting to your web pages, if it is bad, request source to remove, else disavow it.

Your spikes will be improved