In-universe, according to StarTrek.com:

Ensign Pavel Chekov transferred to a new assignment during the final year of the U.S.S. Enterprise's historic five-year mission (2269-70), replaced by his own navigation instructor at Starfleet Academy, Lt. Arex. By stardate 7410.2 (2271), Chekov had received a promotion to the rank of lieutenant, and had rejoined the Enterprise's crew, then under the command of Captain Willard Decker. By stardate 8130.3 (2285), Chekov had risen to the rank of full commander and served aboard the U.S.S. Reliant under the command of Captain Clark Terrell, until the escape of Khan Noonien Singh from Ceti Alpha V (2285) brought Chekov back to Kirk's crew, of which Chekov remained a key part until approximately 9524.0 (2293).

Out of universe, according to the Memory Alpha wiki:

Chekov was absent to cut down on costs of hiring the voice actors, although Walter Koenig penned an episode of the series,"The Infinite Vulcan".

From Curt Danhauser's Guide to Animated Star Trek:

The reason Chekov wasn't on the Animated series was money.

[...] At first, Filmation planned on not having George Takei and Nichelle Nichols come back to do their roles again. But when Leonard Nimoy learned of their exclusion he said that he would "... not be a party to this if two of the minorities who contributed to making STAR TREK what it was when we were on television cannot be incorporated."

It was due to Nimoy's stand that Sulu and Uhura's characters made it into the animated series. Unfortunately with that many star voices, the budget simply didn't allow for Walter Koenig to return as Chekov.

From TVparty!

There were some conflicts on the series. Nichelle Nichols’ and George Takei’s cartoon likenesses were being used but they weren’t originally hired to provide the character’s voices (Paramount owns their younger faces, apparently). Hoping to save money, Roddenberry decided to have Majel Barrett voice Uhura and James Doohan do his best Sulu imitation.

Only a promised work stoppage by Leonard Nimoy got them on board.

Joel Eisner:
"It is common knowledge that Ensign Chekov was cut from the Trek cartoon series for budget reasons. To compensate Walter Koenig, he was allowed to write one of the episodes. However, Chekov was seen in the cartoon pilot episode "Beyond the Farthest Star" in about three scenes which take place on the bridge - you can clearly see for a few seconds Chekov sitting at the helm with Sulu, only to have it cut away and come back with the three legged alien, Arex (voiced by Jimmy Doohan) sitting in his place. I used to have 16mm film print of this episode so I was able to examine it closely, but it is clearly visible in the episode."

Chekov was actually in the first draft of One Of Our Planets Is Missing, written by Marc Daniels. This early version does not feature Arex or M'Ress.

From the way it is written, it almost appears that Stephen Kandel may have created Arex and M'Ress. The first draft of Mudd's Passion describes pilot (not navigator) Arex, almost as if he is a new character, and two 'felinoids', as part of the bridge crew. In the final draft, Kandel names M'Ress, and also describes her as if a new character.