Where can I find the flowers from being 'flowered' by other users? It says they are left at home but I don't see them in my inventory.
You don't actually receive the flowers that other users 'send' you for Vesnali. You get the Vesnali coins from them and they appear in your inventory.

the way to "get" vesnali flowers is to do an easter egg hunt to see if there's one floating on any particular page of Subeta. It's random if there is one. If you do see one, click it and you will get a little windowlet that will tell you that you got it. There's CSS code that will do the hunting for you, and this is allowed. Clicking on the flower-image is the game of it.
You can send people flowers (of all kinds, not just Vesnali flowers) by either clicking on the item and sending it directly, or by going to the Gift Center in Arctic Frost. Doing this will not generate Vesnali Coins.
If you wish to get Vesnali Coins to spend on Esther's shop, or if you want someone to get coins generated for them by your 'act of sending' you specifically need to go to their profile and look for their Option Box (it will have a random flower image) or use the code flower = theirusername (wth brackets, without spaces. The site will then check your Inventory to see if you have any found-flowers and will ask you which ones you want to give. They won't see the flower, so it won't matter what one you pick. They will only see the coins.
If we don't get the flowers that it's referring to why do the flowers keep changing? I would like to hand out more flowers and since I'm not finding many flowers doing quests, I need those flowers. Doesn't make any sense.
why do what flowers keep changing?
On your profile, there is an Option Box, and this Option Box is used by various Subetan holidays. The image on this Option Box is always a random image of something relevant to the holiday? This image of a flower you are seeing in the Option Box on a profile? It's not an item.