What Is Holo-Battle Sim?
Holo-Battle Sim is a roguelike-style solo mode built around Elements. In what we have checked, there are at least 3 Element categories: Grass, Water, and Fire.

Each Element category is split into 4 difficulty tiers, from easiest to hardest: Easy, Normal, Hard, and Challenge. Every tier has its own First Clear Rewards and Training Incentives.

How to Unlock Holo-Battle Sim
Holo-Battle Sim is reportedly unlocked by clearing the level 45 Promotion Assessment.
All 7 Training Styles Explained
Before you start a run, you pick a Training Style. Each style supplements your build with a buff and increases your odds of obtaining a matching Gizmo, and selecting a style also lets you transform into the Aniimo featured in it.

Energy boosts basic skill damage and increases your chance of obtaining Energy Gizmos. Its Core Gizmo is the Energizing Bracer, and selecting it lets you transform into Irisal.

Surge boosts Ultimate damage and increases your chance of obtaining Surge Gizmos. Its Core Gizmo is the Surge Engine, and selecting it lets you transform into Bolty.

Valor increases your damage as you take more damage, and increases your chance of obtaining Valor Gizmos. Its Core Gizmo is the Fervor Mask, and it starts you off with 500 Training Coins.

Vitality grants you a buff when you receive healing, and increases your chance of obtaining Vitality Gizmos. Its Core Gizmo is the Repair Matrix, and it starts you off with 250 Training Coins.

Ironclad boosts your defense in combat and increases your chance of obtaining Ironclad Gizmos. Its Core Gizmo is the Flamewreath Armor, and it starts you off with 250 Training Coins.

Wealth spends your Training Coins to attack, and increases your chance of obtaining Wealth Gizmos. Its Core Gizmo is the Predictive Dice, and it starts you off with 1000 Training Coins.

In what we have checked, the selection screen has 7 Training Styles in total: Energy, Surge, Valor, Vitality, Ironclad, Wealth, and Random.
Which styles you can pick from seems to depend on your account, and it may change daily: two players compared accounts and got different styles, and one saw their own options change from one day to the next. This has not been confirmed over multiple days yet.
Random picks one of the other styles for you, so it is less predictable, but it starts you off with 1000 Training Coins.

How Training Incentives Work
First Clear Rewards can only be claimed once per season, when you clear the corresponding difficulty for the first time.
Training Incentives are different: they refresh weekly, and each Element training has its own set of them.

You do not need to clear every difficulty one by one to collect Training Incentives. Clearing the highest difficulty directly claims all Training Incentive rewards from every lower difficulty as well.
Where to Check Recommended Aniimo
Before you select Start Training, check the Training Details screen for that Element. A Recommended Aniimo button sits right above the Start Training button and shows suggested picks for that specific training.

What Are Gizmos?
Gizmos are passive buffs you pick up while running Holo-Battle Sim. Which ones you see more often depends on the Training Style you picked at the start.

As an example, the Energy Expansion Gizmo (an Energy category Gizmo) increases your max EP by 10.