What does “additional claim” mean?

An additional claim is something you assert beyond what your starting statement itself establishes.

Compare:

Slavery was abolished.

→

People legally classified as slaves ceased to have that legal status.

Very little additional content is introduced.

But:

Slavery was abolished.

The labor market became more mobile.

Now you have added a proposition about labor-market behavior.

And:

Slavery was abolished.

The labor market became more competitive.

That adds another economic proposition.

And:

Slavery was abolished.

Abolition was designed to facilitate capitalist industrialization.

That adds a claim about historical motivation.

So the progression is roughly:

definition → direct consequence → historical consequence → causal interpretation → motive

Each step moves farther from the original proposition.