What's so hard about guest-hosting Jeopardy?

Possibly something to do with the fact that they film 5 episodes a day during the filming period and the presenter having to make themselves look completely fresh for each and every one. It's a punishing schedule and for such a high-profile show, it needs to look perfect every single time.

The Official Jeopardy Website has an article detailing Trebek's working day:

5:15am when he wakes up. After a quick morning routine, he’s out the door at 5:30am sharp and in his truck on the way to Stage 10 to shoot five episodes of Jeopardy! A quick 30-minute commute puts him in Culver City at 6am
6:00 - 7:30am: Reads the newspaper, does the daily crossword puzzle, signs autograph requests and viewer mail.
7:30 - 8:45am: Receives the five "show scripts" for the day from head writer Billy Wisse, and reviews each script, making diacritical marks, looking up pronunciations, identifying potential problem clues, etc.
8:45am: Meet with Rocky Schmidt, Supervising Producer, to go over schedules, personal appearance and publicity requests, etc.
9:00 - 10:00am: Meets with Harry Friedman, Lisa Broffman, Billy Wisse, C&P, and other writers and researchers and go over all five shows, discussing issues anyone may have found in those shows.
10:00 - 10:30am: Relaxes in his dressing room, where the TV is usually tuned to the news or Turner Classic Movies.
10:30 - 11:15am: Hair and makeup.
11:15am - 12:00pm: First show of the day tapes during this time, and the games usually tape true to time, unless there is a technical problem. During the tapings of all shows, in the commercial breaks, Alex has a back-and-forth discussion with the audience, takes questions, and engages in playful banter with the audience.
12:00 - 12:15pm: Changes into a new suit. The returning champion also changes clothes so that people at home don't wonder why they only have one outfit for the week.
12:15 - 1:00pm: Alex tapes the second show for the day.
1:00 - 1:15pm: New show, new suit.
1:15 - 2:00pm: Tapes the third show of the day with the same audience. Each tape day has two audiences – the first audience sees the first three tapings, the second audience sees the last two.
2:00pm: If there are special promotional requests, Public Service Announcements, contestant search promos, and anything else to record, those are done at this point. Then the whole crew breaks for lunch from 2:00 to 3:00, where Alex usually enjoys the soup of the day or sushi.
3:00pm: Make-up/touchup.
3:00 - 3:45pm: Tape fourth show in front of a "new" studio audience.
3:45pm: Last suit of the day.
3:45 - 4:15pm: Tape the last show of the day – the fifth.

So it's not just a case of turning up, chatting a bit, asking a few questions and going home again.

The presenter (and crew) need to do this to a high level of quality (and make it look easy!) every day for 46 days before turning the studio over to other shows.

So yes, this is a tough job for someone who's not used to this schedule and this level of pressure in presenting one of the highest rated shows on TV.

Injecting personality, a few seconds at a time

Jeopardy is a rather fast-paced game, usually squeezing about 60 questions plus some banter into a 23-minute program. The host has to walk a fine line between adding color commentary and keeping the game moving. If the host simply reads questions and answers robotically, they won't project any kind of personality that makes a host likeable. On the other hand, if they say too much, they'll bog the game down with unnecessary conversation, hurting the pace of the show.

A host needs to say enough to showcase their personality and hosting style, but cannot say so much that they take focus away from the players and the game itself (or even reveal game-pertinent information!). The rapid-fire nature of the game makes the format very different from a traditional interview, where there is less time pressure, and less variation in subject matter. Additionally, aside from the contestant interviews, there is little back-and-forth between the host and contestants, meaning that any quips from the host must stand on their own. Most of the game is not conducted in an interview format, so interview skills won't necessarily be useful - rather than trying to hold an interesting conversation, a Jeopardy host for the most part provides one-sided commentary.

Add in the wildly varying subject matter, as well as the quickfire nature of the game, and the host has the difficult task of conveying their personality in very short bursts while staying on-topic in a non-conversational format. A good host will keep the game moving and project a knowledgeable and personable air, but a great host will make it look effortless.

They're not judging the players' answers, there are judges for that. Both Cooper and Whitaker are veteran journalists, they have plenty of experience interviewing people, so chatting with the contestants should be easy for them.

The other answers address your assumption that the host isn't judging answers, but IMHO there's a big flaw in the logic that having "plenty of experience" necessarily means the job "should be easy".

It's certainly true that having experience can make some parts of a job easier, as compared to a person without that experience. But it's definitely not true that just because a person has experience, the job becomes easy.

Indeed, one sign of a true professional is that they put their full effort into the task at hand. Their experience may help guide that effort, and gives them tools to apply that effort more efficiently than would be possible with less experience, but it's still work. It requires mental focus and/or physical labor to do a good job, and especially to do a job worthy of the compensation and expectations given to a person with a lot of experience.

Having a lot of experience doesn't mean being able to coast professionally. It usually means increased competence, but hard jobs can, and usually do, remain hard.

I tried out for Jeopardy while stationed in Iceland in 1993. The USO brought Alex Trebek to participate in this, and he answered a number of questions for us.

I don't know if he did this up until the very end, but he also knew the correct questions to all of the answers when he did the show. I don't know if he had to cram or if he was simply a walking encyclopedia.

From what I could tell he was a class act all around.