start_incremental is not a call for a timed indexing of the catalog. It simply starts an incremental population for fulltext_catalog_name based on the last time stamp of the last index to the current time of running the start_incremental. It takes the last timestamped index (there must be a column in the table that has a timestamp) and then indexes everything betwee the two times. It is different from a full index as it only updates that which it is missing instead of the entire table.
There must be something calling the procedure, whether it is a job or some kind of trigger within the software that causes it. You'll have to contact the software vendor to find out more information.
The command is being run every minute by a SQL job. I assumed an incremental would be run perhaps once or twice a day and not every minute. What advantage is there to run this once a minute and is it possible to suppress the message seen in the log?
I don't believe there is a way to supress it and you may not want to in case the index fails and the software vendor needs the information the log may provide to them. AFAIK, there is no way to supress indexing logs.
At a previous job, I know that Arcserve's database engine queries and writes every minute and their tech support was adamant about keeping those going. It will just depend on your vendor. I could see the advantage in updating the indexes incrementally if there is a large amount of data being written by the minute, such as in backups. Every file would be cataloged so that you could recall it easily from the backup engine and restore the file quickly. I suppose it depends on the software, but I would suggest contacting the software vendor for that information. There could be a wide range of reasons and I am only speculating based on prior experience.