The Emperor was very, very paranoid about the status of House Corrino.
He was the emperor, yes, but Leto was his cousin, also of noble birth, and Leto was vastly more popular than anyone else at the time in the Landsraad.
Leto was the unofficial spokesman of the Landsraad, in fact, their "highest position".
That alone sent Shaddam into fits, someone who was able to claim the throne by virtue of noble blood was popular and well supported among most of the noble houses.
However, Leto also had a huge private army, and, House Atreides developed training that made their soldiers superior to the Sardaukar.
"The Padishah Emperor turned against House Atreides because the Duke's Warmasters Gurney Halleck and Duncan Idaho had trained a fighting force -- a small fighting force -- to within a hair as good as the Sardaukar.
Some of them were even better.
And the Duke was in a position to enlarge his force, to make it every bit as strong as the Emperor's.
"
This is a quote from Thufir Hawat.
If the Atreides had managed to train their entire army in this manner, they would have had a force bigger than the army of House Corrino that was also better in combat than the Sardaukar.
I assert that they would have been better because they were clever where the Sardaukar were brutal.
Shaddam IV knew all of that when he set in motion the plan to lure the Atreides to Dune and use their age old vendetta with the Harkonnens to destroy them.
You see, your question asks why the Emperor agreed, but, really it was his plan.
Hence, "plots within plots".
He had plotted all of this and made the Harkonnens think it was their own idea!
As to "why does it have to be on Arrakis but not Caladan"?
If you're asking why the emperor or the Baron can't attack until after the Atreides leave Caladan, it's because Caladan is a huge planet with billions of people and wantonly attacking the fiefdom of a noble house with billions of witnesses is certain to bring censure from the Landsraad.
On Dune, the only real witnesses who aren't Atreides are the Fremen, and they aren't going to say anything.
scifi.stackexchange.com/users/5828/Nathan C. Tresch
The Emperor was suspicious of the Duke Leto and wanted him gone.
He could not move against him directly as the Lansrad should not approve due to the popularity of Duke Leto.
So, the Emperor used the vendetta between Baron Harkonnen and Duke Leto as a tool to hide his true motives.
This had a dual purpose:
destroy House Atreides and put House Harkonnen in the emperor's debt for a long long time.
Keep reading the book, it is all explained therein.
scifi.stackexchange.com/users/2682/Sardathrion - against SE abuse
The subliminal part of the plan and true goal... Economics.
Leto: "Imagine what would happen if something should reduce spice production.
"Whoever had stockpiled mélange could make a killing."
Paul said.
"others would be out in the cold" ...He nodded. "
The Harkonnens have been stockpiling for more than 20 years.
"
"They mean Spice production to fail and you to be blamed." "...
Knowing that the Harkonnens stockpile mélange, we ask another question: Who else is stockpiling?
That is the list of our enemies.
Certain houses we Knew were unfriendly and some we'd thought friendly.
We'd need not consider them for the moment because there is one other much more important: Our beloved Padishah Emperor."
The Emperor is the only person authorized by law to hold spice stockpiles.
A major reason for the Emperor's unpopularity is that he launched several military operations against Landsraad nobles in possession of illegal stockpiles.
One such raid used forbidden atomics and provoked legal backlash against the Emperor from the Landsraad.
The basic economic principle of supply and demand dictates that if supply is reduced the value increases.
House Corino's wealth and thus power would be greatly improved by a spice production bottleneck.
Ridding himself of the popular Duke Leto is just a bonus.
scifi.stackexchange.com/users/70926/RedOculus
Arrakis is so remote and relatively uninhabited that the Emperor could do as he wished there
(ie: assassinate Duke Leto), and then write the "history" of the event to suit his own purposes.
House Atreides was forced into moving to Dune because they could not refuse Shaddam's request without appearing either foolish or disloyal or both.
House Atreides fully understood the defensive situation that they were put in by the Emperor (Hawat was no fool), and the first book begins with this given assumption.
The tenets of Kanly were always closely followed, as a matter of honour amongst the Landsraad.
Shaddam is trusting that he can discredit or destroy House Atreides with his "magnanimous" decision.
Leto is acting very boldly by taking an even greater position of power offered by one who is surely uncomfortable with the level of power that the Atreides already exhibit.
scifi.stackexchange.com/users/16311/T Chalmers F
Shaddam was also recovering from the amal scandal that occurred 15 years prior to the events in book one Dune (read the House Chronicles, house Atreides, house Harkkonnen, house Corrino, in that order).
Shaddam was placed under sanctions for his role in the debacle that was exposed by Duke Leto's attack on the Tlelaxu who had conquered Ix 35 years prior to the book Dune .
Duke Leto was also the one who suggested sanctions against Shaddam instead of outright removal from the throne, in order to ensure that the imperium did not fall into a massive civil war by members of the Landsraad trying to claim the throne for their selves.
Due to all of these events where Leto was perceived by the emperor as an enemy, and the fact that Shaddam had no sons but was not sterilized by his bene Gesserit wife.
Accompanied by Leto's rising popularity, Shaddam was not only feeling threatened, he was guaranteed to ceed the throne to some one not bearing the Corrino name and he didn't want his nemesis cousin Leto to be the one to marry Iralan and become the next emperor.
Thus, by tricking the Landsraad into thinking the Harkkonen were jealous of the CHOAM contract being taken from them two decades earlier than expected, and given to their arch rivals the Atreides and the attack was simply spite, Shaddam kept his already sanctioned butt from falling back into the hot seat.
scifi.stackexchange.com/users/56661/Joseph mastrangelo
I am reading book 1 of the Dune series. I just read the chapter in which the Harkonnen took control over the house of Atreides in Arrakis with the help of Yueh.  I know the Emperor sent a legion of Sardaukar (disguised as Harkonnen troops) to help the Baron Harkonnen to kill Duke Leto and his men. Why does the Emperor agree to do so? And why does it have to be on Arrakis but not Caladan?
Why did the emperor send the Atreides to Arrakis, then the Sardaukar to remove them?
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scifi.stackexchange.com/questions/15057
scifi.stackexchange.com/users/1801/lamwaiman1988
The Emperor was very, very paranoid about the status of House Corrino.  The Emperor was suspicious of the Duke Leto and wanted him gone. The Emperor was paranoid about the status of House Corrino. Moreover, the Emperor was suspicious of the Duke Leto.
The Emperor was paranoid about the status of House Corrino. The Emperor was suspicious of the Duke Leto.