To cross-apply an argument in debate is to take an argument made earlier in response to one issue and use it as a response to a different issue in the same round. Cross-application is efficient - it lets a debater respond to more of their opponent's arguments in less time without having to make new points from scratch.
Example
If the Opposition argues that the Proposition's economic impact is uncertain, and the Proposition earlier established strong economic evidence for a different contention, the Proposition can cross-apply that evidence to respond to the uncertainty claim. 'Cross-apply our evidence from contention one here - that same evidence shows the impact is certain, not uncertain.'
Why Judges Like Cross-Application
Judges appreciate cross-application because it shows the debater is thinking about the round as a connected whole rather than as separate arguments. It also saves time, which allows speakers to cover more ground within their speech. A skilled debater might cross-apply three or four arguments in a single speech.