Why this is happening?
This is happening (showing the tags list page on homepage) because there is no menu item for this particular view. So Joomla, because it needs a menu item for each component's view, will use the default (homepage) menu item to display the relative content of the tags component.
Obviously your homepage's menu item has assigned on it many modules that are positioned above the component output, and likely in its standard condition is set to not display Joomla content at all - e.g. a featured articles menu item that is set to display 0 articles - or maybe you have a single article menu item, which shows your home article below these modules.
But when you click on any of the tags, the module is linking to a view of the tags component - and without a dedicated menu item for that view it will default to the homepage. So at the moment, your request will be slightly different than when you actually see your real homepage. If you had disableb SEF URLs you would see in that URL something like:
with the itemid being the id of your home page, which will force to load all modules of the homepage at their current module positions on the template, but this time mixed with the content of the tags component.
To overcome this, you simply need to create a menu item for that specific view of your tags component. If the module is well designed, then normally when you click on any of the tags, Joomla will find a dedicated menu item to display your tagged items list layout on its own and not under the homepage's menu item. You will be able to tell to Joomla what modules you will like to display for that view as well.
Hidden Menu Item
If you don't want to show that new menu item for the tags under any of your visible menus - then you can create a hidden menu/menu item.
Helpful resources for further reading: