"A Mighty Powerhouse of Strength! THE FEARSOME... LEAF!"
Red's Venusaur has the same name as the default name for the female trainer from Fire Red/Leaf Green. "Leaf is male" as a reference to Red being transgender.
In-universe, Red named him that because it literally grows leaves.
Leaf has been around humans his entire life, hatched as an egg in Professor Oak's Pokemon lab, raised alongside Charmander and Squirtle as its siblings.
The Bulbasaur had ample space to play in and interact with the other Pokemon that Professor Oak took care of. Though the fields were vast and it had all the friendship it could want, he always had grander ideas for adventure. Often, it would sit in a special place on a hill with a flower bed and look out to the distance. He could see trees, a forest, and a far distant mountain. In it's mind, it wondered what those places would be like if he was actually there.
Red (known as "Dawn" at the time) and Blue would read together and play get kid games and act like pals, but when they disagreed (which happened most of the time) they'd fight. It was less of bullying and more like "I'm smarter" then trying to prove it.
When they both got interested in Pokemon, they were always trying to one up each other with who knew the most. They would consume every book they could get their hands on, which was a lot because professor oak was more than happy to give them some.
Professor oak saw potential in them. They were both very driven, determined, and eager. When he suggested the idea to them that they could be Pokemon trainers, things got even more tense. Their arguments started to shift from just a knowledge of facts, but rather what sort of pokemon they would catch.
Professor Oak wanted to give the them choice between Bulbasaur, Charmander, and Squirtle to choose as their first Pokemon. Professor Oak told them the plan, and they were given a month to carefully consider their choice.
As expected, throughout this month Red and Blue would hotly debate which one would be the best.
Charmander was a Fire Type. Fire was powerful, and fire pokemon were very difficult to find in the wild and hard to capture. Having one handed to you would give you an immediate competitive edge, especially where they lived where many of the nearby wild pokemon were weak to fire. It's evolved form, Charizard, could fly you from place to place easily. It is said they are difficult to train, but these two were confident in their abilities.
Squirtle was a water type. A strong one at that. They're known to be amiable, and as water types they could learn important moves and surf you across the oceans if properly trained. It's evolved form Blastoise was known to be quite ferocious.
Red would eventually are you that Bulbasaur was the best, though. It's status moves could be dehabilitating, it had a type advantage over water to capture one easily, and it's evolved form was said to have powers over nature.
When it came time to choose a starter, Red's mind was already made up. He picked Bulbasaur, named it Leaf, and since then, they've been inseparable.
As a Venusaur, Leaf can control the nature around it. He's so heavy that a single stomp of its feet can cause an earthquake. He is usually stationary, feeding on the sunlight, feeling the earth, and listening to the plants around it unless it is actively traveling. Instead, it will often grab things using the vines on its back.
Red will sometimes sit on his back so Leaf can carry him instead of walking himself. Leaf has no problems with the extra weight, and will do so without issue.
If you have no senses to detect its strong magical power and force over nature like a Pokemon, you will still be intimidated by its large size and fearsome presence.
The flower on the back of every Venusaur dies once a year and regrows as part of its natural life cycle. When it regrew on the mountain, it regrew as a large Hellbore flower. Hellbore grows in snowy climates means serenity or anxiety. It symbolizes the comfort and wisdom Red has found on the mountain, as well as the fear keeping him up there.
Leaf was in charge of maintaining a garden for the rest of the Pokemon to eat from. He grows the plants, and it powers itself using sunlight that Sunny the Arcanine provides by using the move sunny day.
For food, aside from sunlight, using its vines it will pull dead trees out of the ground roots included and crunch them in its massive jaw. Leaving nothing but splinters behind. The tree easily crunches to its strength of its jaw. After eating, it will use the leftover sticks to play fetch with Sunny. Leaf also eats birds sometimes.
Leaf lives with Red and Blue in their Pokemon Gym. He is far too powerful to use against trainers who challenge the gym, so it instead gets to spend most of its time relaxing sunbathing outdoors. Leaf spends most of his time with Blue's Charizard, as they consider each other's siblings. one second
Red's Cherrim admires him greatly, and will often ask for advice from the fellow Grass Type.