It is alluded to in episode IV, and explicitly stated in the EU literature, that Han Solo incurred his debts to Jabba the Hutt by dumping a significant amount of cargo whilst being inspected by the Empire.
Why did Han dump the cargo overboard instead of storing the contraband in the Falcon's hidden compartment that we see being used to hide Han, Luke, Chewbacca, etc. in A New Hope)?
This was addressed in the (now Legends) EU novel Rebel Dawn. In short, the barrels of spice were already inside the smuggling compartments when Han and Chewie detected an Imperial ship bearing down on them. Han knows that the compartments are sufficiently shielded to pass inspection by bored customs officials but they won't stand up to the scrutiny of an Imperial 'Scanning Crew' on a mission to hunt down Spice smugglers. Han decides to dump the cargo, taking care to log its direction of travel so he can collect it later.
“Chewie, we ain’t gonna be able to outrun these Imps,” Han said. “And we’re outgunned. But we’ve lost that guy on our tail, at least for the moment. I think if we can get far enough ahead, we should go ahead and dump the load just inside the Pit—the way you did that time with Colonel Quirt on that other Run. After they’ve searched the Falcon to their hearts content, we come back and retrieve the cargo.
and
With luck, Han should be able to track the spice’s progress and find it again after a little searching. The barrels themselves were made of an alloy that would show up on his sensors if he got close enough.
and
“Captain Solo, you stand in suspicion of smuggling spice from Kessel.
I am authorized to search your ship.”
Han waved at the interior. “Search away,” he said. “I got nothin’ to hide.”
Capucot sniffed and managed to stare down his nose at Han—even though the Imperial officer was several centimeters shorter than the Corellian.
The captain beckoned a scanning crew into the ship. “Search every millimeter, “he ordered. “I want that spice.”
Han shrugged and stepped aside.
The Imperials searched … and searched … and searched some more.
Han and Chewie winced as they heard crashes from the lounge and the aft cargo compartment. “Hey!” Han protested, “I’m just an honest trader!"
Whether these details will remain the same when the new stand-alone Solo movie comes out (or whether this incident will even be addressed) remains to be seen.
No Disney canon info I'm aware of.
In EU canon, there were two reasons:
This was a standard tactic used by him successfully before
It was far less risky (yes, a hidden compartment is hard to detect, but not impossible).
Han’s mind raced. “Chewie, we ain’t gonna be able to outrun these Imps,” Han said. “And we’re outgunned. But we’ve lost that guy on our tail, at least for the moment. I think if we can get far enough ahead, we should go ahead and dump the load just inside the Pit—the way you did that time with Colonel Quirt on that other Run. After they’ve searched the Falcon to their hearts content, we come back and retrieve the cargo. Whaddaya say?”
("Rebel Dawn" (Han Solo Trilogy #3 by A.C. Crispin), Chapter 15: The Last Kessel Run)
They were sure Imperials would be extra careful searching since they seemed to have been staging a deliberate ambush for the Falcon:
Chewie suddenly started whining and gesturing frantically at the board. Han looked at his instruments and groaned aloud. “Oh, blast! Three Imps out there on the perimeter of the Pit! What else could they be doin’ but waitin’ for us! And one of ’em is a big sucker!”
I believe you are looking for:
Darth Vader: "Send a Scanning Crew onboard, I want every part of this ship checked"
Given that the officer reporting to Vader reported that the ship's logs and his walkthrough indicated there was no-one onboard, a thorough scan was ordered, perhaps something you'd do as well if you were patrolling for smugglers and caught a possible one in your net.
And it turns out Han had an idea of what comes after the initial boarding, given that they 'subdued' the crew and the Stormtroopers that came after
