When you visit the bank, scans your items and deposits all which exist in your inventory and your bank and which stacks in the bank have space in them.

Does not add things that aren't already in your bank.
Does not add new stacks of things that are already in your bank.
Does automatically stuff the stacks you have in your bank until they are full.

Examples (to clarify functionality):

Player has 4 stacks of rawhide in their bags for some reason they have been split into stacks of 15 each
Player has 2 stacks of rawhide in their bank, one of 75 and one of 85
Upon opening the bank the 85 stack will get 15 making a stack of 100 and the 75 will get 25 making another stack of 100
The player will be left with one stack of 20 rawhide in their inventory and have 200 rawhide in the bank

Another player has one stack of 40 rawhide in their inventory
Player has one stack of 40 in their bank
Upon opening the bank, the player will have no rawhide in their inventory and one stack of 80 in their bank