"This House" is a prefix used in the construction of resolutions. Often phrased as "This House Believes That...", the phrase is basic terminology when it comes to forming resolutions.
"This House" as a phrase for debating came from Parliament, where the House of either Lords or Commons debated an issue. When a vote was reached into a majority, the entire house via collective responsibility supported the conclusion made. Hence "This House". The principle naturally transfers to debating. Either the motion passes and this house supports the argument, or this house rejects the motion.
One debater named Jedi4 even parodied this resolution term to "this gay bathhouse" in one particular debate.