Log file increases while differential backup is in progress
Now, the problem which started recently is whenever our scheduled History database's differential backup is progress log file of History database increases to 10+ GB and as soon as it completes everything gets normal.


So was transaction log backup also running before the differential backup job? Starting from SQL Server 2005 both Differential backup and log backup can be started simultaneously on SQL Server but transaction log truncation would be delayed until the diff backup completes that is why till differential backup is running the log file grows to write transaction after it completes the logs are truncated and space is reused. In your case Log backup would have run while differential backup was running and would not have been able to truncate logs but as soon as it finished the logs were truncated

Full backup and differential backup can delay log truncation because both needs to read some part of log to create consistent backup and if log backup was allowed to truncate the log file this would result inconsistent full and differential backup 

This is similar SE thread I answered before.