Why doesn't Zorn's lemma apply to $[0,1)$?

Consider the real interval $[0,1)$, this is partially ordered set (totally ordered actually). This set has a upper bound like $1$, and according to Zorn's Lemma each partially ordered set with a upper bound should have at least one maximal element. However, in this set there is no maximal element, i.e., element that is greater than every element of the set because you can be as close as to $1$. I am should I don't understand Zorn's Lemma. Please Help!!

As a partial order $[0,1)$ has no upper bound. Sure $1$ is an upper bound of $[0,1)$ in $[0,1]$ or in $\Bbb R$. But that is not the same partial order. You are not allowed to go to larger partial orders when you apply Zorn's lemma.

So $[0,1)$ has many chains without upper bounds. E.g. $[0,1)$ itself.

You are actually not using Zorn's Lemma which states that if each chain in $[0,1)$ has an upper bound, then $[0,1)$ has a maximal element. However, the chain $\{1- \frac 1 n \}_{n \in \mathbb N}$ has no upper bound in $[0,1)$.

To apply Zorn to a linear order $L$ is pointless: in order the prove the existence of a maximal element (here: also a maximum), you first have to give a maximum for $L$ as the condition says we must have an upper bound for every chain in $L$, including $L$ itself. The conclusion would be weaker than what you need to show in the proof anyway.