What does depleting the break gauge do?

The damage half of that is the part people build around. A staggered enemy takes significantly more from everything, so the break window is where your cooldowns belong. Chipping away with your biggest hits while a shield is still up wastes them. Fill the gauge first, then dump.

The other half is defensive and easier to overlook. Because the break interrupts whatever the enemy was in the middle of, the gauge doubles as a panic button against attacks you cannot outrun. If you can empty it fast enough, the attack simply does not happen. That makes break pressure worth carrying even in fights where you are not short on damage.

Why early teams carry a break unit

A workable starting composition is a DPS unit, a break unit, and a healer or support. The break unit exists to open the window and the DPS unit exists to use it, which is why the pair is treated as one decision rather than two separate picks.

Do not choose that break unit off its role label alone. Labels in Aniimo are assigned, but what a creature actually does comes down to its individual skills, to the point that creatures not labelled as healers can heal. Read the skills. A support in the third slot compounds the window further, since support buffs amplify outgoing damage on top of the amplification the break already applied.

Breaking before you throw a pod

Breaking a wild Aniimo also improves your chance of catching it, which stacks with the general rule that weakening a target in battle raises its capture chance. A break is not wasted effort on a creature you intend to keep.

The rest of the catch maths is covered in the linked article on catch stance and combat roles. What carries over here is that breaking is a catch tool as much as a damage tool, and that the displayed catch percentage can still fail after the pod shakes, so raising that number before you throw is the only lever you have.

Not settled: How the amplification scales is not pinned down. 200% is the figure attached to a fully emptied gauge, but whether partial break pressure gives anything less, or whether some enemies resist it, is not settled.