Why is A union B also called "A or B"?

Because the elements of $A \cup B$ are exactly those objects that belong to $A$ or belong to $B$.

The claim "$p$ or $q$" is true if either $p$ is true, or $q$ is true, or $p$ and $q$ are both true.

That is, the claim "I am human or I am over one meter tall" is true, because at least one of the subclaims is true (in this case, both are true)

Suppose we are in a university.

The President orders ( for some reason we do not care about) the following thing: I want all math students and all students that play tennis to be present in the great hall tomorrow ( and no other person).

The next day, due to the strong authority of the President, all math students and all students playing tennis are present in the great hall.

Certainly, ***the President has in front of him all the members of the set : $M \cup T$ ( that is the union of set M and of set T)

with M = the set of all math students and T = the set of students playing tennis.

Now, what does the President know of any one of the students present there ( although he has possibly never met anyone of these students before)?

For example, what does the President know about David, one of the students that is there?

The only thing he knows for sure is that :

David is a math student OR David is a student that plays tennis.

Remark : of course, it could be the case David to be both a math student and a tennis player; but of this, the President has absolutely not idea.

In other words he only knows for sure that : one at least of these two propositions is true :

(1) David is a math student

(2) David is a student that plays tennis

and this " at least one" is precisely what defines the inclusive OR operator in logic.

Since the same thing could be said about any student, one can say in general that :

The set : M U T is the set of all x such that (1) x is a math student OR (2) x is a student that plays tennis.

In symbols : $M \cup T$ = { $x$ | $x$ belongs to $M$ $\lor$ x belongs to $T$ }

You might say : but there are certainly students in the hall that both are maths students and play tennis.

You would be right, but notice that if a student, say, David is both a math student and a tennis player, this is precisely an excellent reason to say that he is at east one of these two things. So David belongs to the set M U T exactly in the same way as all students that are at least one of these two things : math student OR tennis player.

The OR that defines the union operation is inclusive.