The burden of proof in debate is the responsibility to prove something - to establish a claim with enough evidence and reasoning that a reasonable judge would accept it. In most formats, the side proposing change carries the primary burden of proof because they are asking the judge to move away from the status quo.
Who Has the Burden
In a policy motion, the Proposition has the burden to show that their proposed policy is better than what currently exists. The Opposition can often win simply by showing that the Proposition did not meet that burden - they do not necessarily need to prove the status quo is perfect. This asymmetry is important because it means the Proposition usually needs to do more work to win.
Burden in Value Debates
In value debates (where the motion is not a policy but a comparison of ideas), the burden is usually more balanced. Both sides share the responsibility to prove their claim is more true or more important than the opposing view. In those rounds, framework becomes especially important because it tells the judge how to decide who met the burden better.