When they flower is when they start to set potatoes to grow. When the vines start to die they can be harvested, though they might store better if you wait until the vines are fairly dead before you harvest them. I'm in a hurry this morning so I'll let someone else tell you about groveling for new potatoes.
Some types of potatoes seem to set potatoes all along the stem, so the deeper you bury them the more potatoes you might have. Some apparently don't. I'm getting this from different photos different people have posted on here and their comments.
You can continue "hilling them up" until they start to flower, then it doesn't do you any more good. Don't totally bury the plant but leave a few leaves and buds showing, just raise the level as they grow. The potato plant will continue to grow. The potatoes set on above the seed potato so you need to put some dirt up there to give the potatoes room to grow.
If the sun hits potatoes they will turn green and contain solanine, which can make you very sick. So keep dirt over the developing potatoes. I plant mine in the ground and pull dirt up around them. Sometimes that dirt will crack and expose the developing potatoes so I pull a little more dirt up around them even though they have flowered.
Potatoes re a pretty low stress plant and can be pretty productive but they do need a little work. Mulching is good but after a certain point they pretty much self-mulch.
Don't put a whole lot of faith in that 55 day to harvest time. That's just something that they put out for comparison's sake between varieties. How long it actually takes will depend on your growing season. It probably won't be any less but it could be a lot longer.
Enjoy them and like everyone else, I'm envious of that photo.
