Start amount would be the amount of the product that is already available at the industry. (Lumber mill starts off with 28 logs on hand by default, and can only hold 41 max. If it reaches max capacity, you won't be able to unload logs till some are used up.)
Rate would be the amount of the product used per cycle. (In the case of the lumber mill, the default rate for logs should be to consume 1 log per minute.) Any rates set at 0 would mean that industry would not use/produce that item.
Also, keep in mind there are 2 tracks, make sure the train is going to the log drop off track to unload, and of course make sure you're giving it the unload command.