It means each component is an unknot, but the entire link is not necessarily trivial.
If you take the Lickorish-Wallace theorem for granted (every 3-manifold is obtained by integral surgery on a link in $S^3$), it is not too hard to see how to make every component of such a link an unknot using Kirby moves. You can blow up and then do handle slides to change the crossings of any component of your link without changing the resulting $3$-manifold. It is a classical fact that every knot can be turned into the unknot by changing some of its crossings. Hence we can always alter a framed link in $S^3$ in such a way that we get a new framed link with all components unknots.
Here's a picture of the "Kirby move" that changes crossings (from Gompf and Stipsicz's $4$-Manifolds and Kirby Calculus):
If $M$ is a $3$-manifold obtained by surgery on a framed unknot, then $M$ is $S^3$, $S^2 \times S^1$, or a lens space. If $M$ is obtained by surgery on a framed unlink, then $M$ is a connected sum of copies of $S^3$, $S^2 \times S^1$, and lens spaces. There are many $3$-manifolds that are not of this type. One example is the Poincaré homology sphere $\Sigma(2,3,5)$. Hence in general a $3$-manifold can only be obtained by integral surgery on a nontrivial link with all components unknots, not necessarily an unlink.