I've seen The Matrix when it first came out and now I wish to watch all the Matrix movies together in order of relevance including the Animatrix anime.
Does Animatrix somehow fit into The Matrix world that is depicted within the movies and are there any spoilers in the Animatrix that could reveal parts of the The Matrix Reloaded and The Matrix Revolutions?
The Animatrix has both prequel stories, as well as concurrent stories to the Matrix. One of the stories even becomes explicitly canon when the protagonist of it becomes a minor but important character in Revolutions.
Kid from Kid's Story shows up in Reloaded, but becomes important in Revolutions, when he helps save people during the Battle of Zion.
The story itself takes place between The Matrix and Reloaded.
There is no spoilers for Reloaded or Revolutions in any of the 9 short films.
The film is a compilation of nine animated short films, including four written by the Wachowskis. It details the backstory of the Matrix universe, including the original war between man and machines which led to the creation of the Matrix.
The Animatrix provides a deeper background in the world of the Matrix. It shows how other people experience the Matrix.
I don't think there are any (major) spoilers. There are definitely references though.
The Animatrix will definitely improve your understanding of how everything is tied together.
The Animatrix serves as a canon source of information about the Matrix World. It contains history of the Matrix - the rise of machines, the war, and why the Matrix came to be created, and humanity enslaved. The others detail the world of humans - both in Matrix and outside. Some show how some humans reject the matrix at a subconscious level - and how the machines try to continue to enslave them. Some others show how the Matrix - like any computer program - has glitches - and the effects in world. This is similar to how Neo sees a cat walk twice within the same door frame - this was a glitch - or a bug. Others show the lives of Zionists as they train, and the challenges they face when they realise the world outside the Matrix. The Animatrix serves to complement the Matrix, along with the video games which progress the story further after the films.
Let’s take a step back into what really happened in the world in the
early 21st century. This information is from the series called The
Animatrix which takes a deep dive into what really happened. Do watch
The Second Renaissance Part 1 & 2. These short animated films were
released much after The Matrix. A synopsis from this is important to
know as it is very consistent with the movie(s).
In addition to the other answers, the two mentioned stories from the Animatrix, are two very important prequels that sets the stage for The Matrix. It tells us about how the AI was born. What caused the war between Man and Machine. Why the sky was scorched. And most importantly, why the Matrix was created.
There is a continuous plot thread between "Final Flight of the Osiris", the game Enter the Matrix, and the film Matrix: Revolutions. The girl in FFotO is delivering a message (about the coming Sentinels attack) to a postbox in the Matrix. In the game EtM, the characters have received this message and are relaying it to the captains of all the hovercraft to call them to a meeting. And in M:R they are attending that meeting.