Basically, the word "very" means "same." It usually adds some kind of emotional emphasis to the sentence. "On the very day that X happened" means "on the same day that X happened." "That very day" means "the same day [that we're already talking about]." It refers to a day that was already mentioned previously. "This very day" usually means "the same day [that is happening right now]" -- in other words, "today." If today is May 26, 2021, then "It happened one hundred years ago [on] this very day" means it happened on May 26, 1921. Saying "this very day" instead of just "today" can be used for emotional effect. (The kind of emotion it adds will depend on the context.) "To this very day" is a phrase. We use it to show that something has continued for a long time, all the way up until the present day (today). "It happened when I was five years old, and to this very day, I still remember it well" means that I've continued to remember it ever since it happened, and even today, I can still remember it clearly, even though a long time has passed. The word "very" adds emotional emphasis to the idea that something has remained the same and never changed.

The very day -- not something you would say This very day -- means today That very day -- refers to a different day which should be made clear from the context To this very day -- means until today The use of "very" is emphasising this or that exact day. You would use it to be emphatic when referring to a day. Examples: That very day, as the old man died, the clock stopped and hasn't worked since. But if you go to that house at night, they say you can still hear his footsteps on the floorboards as he wanders restlessly, to this very day. Hopefully you can see I was trying to sound a little dramatic with my two sentence ghost story, but that's the kind of emphasis you might use it for.