The reason this is not "obvious" and instead it is a rather deep mathematical truth is that partial orders, even ones where every chain has an upper bound, can get pretty hairy and intangible, mathematically speaking.
When we think, and try to visualize structure in our heads, and a partial order is structure, we are limited by our imagination, which is often good at modeling finite objects, sometimes large enough that we think of them as infinite. But the mathematical truth is different.
Just as well, it is not obvious that every partial order has a maximal chain. If it was obvious, then indeed Zorn's lemma was obvious. But the fact remains that there are partial orders in the universe of sets which are wild beyond your imagination -- and then some.
Now, to your question, the assumption just says that a chain has an upper bound, which is an element of the partial order which is larger than the members of the chain. A maximal element, however, is an element which no point is strictly above it. Consider the interval $[-1,1]$ in the real numbers, the set of negative numbers has an upper bound, $0$. But this is not a maximal element.
So Zorn's lemma tells us that if every chain has some upper bound, then there is a maximal element. And the chains will gradually become longer, stranger, and at some point they will exceed your imagination.