The TOEFL website today has published a new free speaking task today.

If you use the present perfect tense, you don't use specific time adverbs like "today." If you really want to use it, you can say "The TOEFL website has published a new free speaking task."

InFor this task I had to listen to a conversation between two students about rebuilding an old auditory inon the university campus.

Man hasA man expressed his opinion very clearly about this plan of the university's administration very clearly.

If you say "man" without an article it sounds like you're talking about all of mankind
"Very clearly" can go at the end, but it's more readable if it's closer to the verb it describes

First, the man says the administration wshould not destroy this auditory in because of its bigrich historical heritage in life ofas a part of the university, alsond because its interior is one of the oldest ones.

I don't think "would not" is right here because that's saying they won't destroy it even though it's previously stated that they plan to.
We don't really use "big" to describe heritage. Not really sure what "in life of university" is supposed to mean Dx

Second, he says there are no reasons to make this auditory bigger because he has never even saween this auditory full of students body.

Or "there is no reason" which emphasizes that there is not even a single reason

This makes me tode me understand that this man really likes the university inat which he is studying,; he likes to go ton lectures and notices the details of its lifehistory(?).

I'm assuming you meant "go to lectures" like a student going to class, not that the man likes to give lectures?

The TOEFL website today has published a new free speaking task (/exercise) today.

In this task (/exercise), I had to listen to a conversation between two students about rebuilding an old auditory iium on university campus.

Man hasA man expressed his opinion about this plan ofe university's administration's plan very clearly.

I was asked to retell the main idea (/key points) of his opinion and why he holds it.

First, the man saysid that the administration would not be destroying thise auditory in case of its big historical heritage in life of university, alsoium as it's a large part of the university's historical heritage, and its interior is also one of the oldest ones.

I think he really enjoy itslikes its (/the auditorium's) interior and he will be sad without it.

Secondly, he says there are no reasons to make this auditoryium bigger because he's never seven saw this auditory full of students bodthe auditorium at full capacity.

This makes me to understand that this man really likes the university in which he is studying, he at, - he also likes to go ton lectures and notice (/pay attention to) the details of hits life.