Complex sentence: Under current Hall of Fame rules, they are now awaiting eligibility for consideration by the since-created Today's Game Committee, which is automatic once a player is no longer eligible for consideration by the BBWAA.
Split into simpler sentences: Under current Hall of Fame rules, they are now awaiting eligibility for consideration by the since-created Today's Game Committee. Eligibility for consideration by that committee is automatic once a player is no longer eligible for consideration by the BBWAA.