What does this Apache SOLR error mean?
The error means exactly what it says:


  Request failed: request timed out


The connection between your Drupal server and Solr server timed out.

If the servers are geographically far from one another, you might consider moving them closer together to reduce latency. Your Solr or Drupal server may also need more resources. 

Both are things you'll need to incrementally test (preferably in an identical staging environment) until you've worked out where the bottleneck is and how you can overcome it