An API is a defined way for one system to ask another for data, or to make it do something, with no human in between. For a screen or video platform it is what turns an isolated tool into part of an operation.
The cases that matter are unglamorous. Pushing today's prices from the till system rather than retyping them. Creating a personalised video automatically when a CRM record changes. Reporting screen health into the monitoring dashboard the IT team already watches. Each removes a recurring manual step, which is where the actual saving lives.
The caveat is that "we have an API" says nothing about whether you can use it. The questions that decide are which operations it exposes (read-only reporting is common, full content publishing much less so), whether it is documented well enough to build against without vendor calls, what the rate limits are, and whether it is versioned so an upgrade does not silently break your integration.
Ask for the documentation before signing, not after. A vendor unwilling to share API docs during evaluation is telling you something about what is in them.