If an API is the socket a product exposes so other software can connect to it, the SDK is the toolbox that makes using that socket easy. It is a bundle of pre-built pieces, documentation and examples that saves developers from building the connection from scratch.
An analogy: the API is the rule that says "our power sockets look like this." The SDK is the box of compatible plugs, adapters and instructions the company hands you so you can wire things up quickly.
Why you keep hearing it
Because AI companies release SDKs to get people building on their models, and those releases make the news. When Anthropic ships an "Agent SDK", it means: here is a toolkit for building your own AI agents on top of our technology. The existence of an SDK is a signal that a company wants an ecosystem around its product, not just users.
What it means for you
You will never use an SDK yourself, and that is fine. Read it as a signal. "We offer an SDK" translates to "developers can build products on top of us." If a platform you rely on ships one, expect more tools and integrations around it soon.