Is there any motivation for Zorn's Lemma? I have been reading Kreyszig's book on functional analysis, where it uses Zorn's lemma to prove the Hahn Banach theorem. However I don't quite get what Zorn's lemma is saying. 
I understand that it is an axiom and it is equivalent to the axiom of choice, but axiom of choice seems much more intuitive to me. So is there any way to understand the Zorn's lemma in a more intuitive way?

I had similar conceptual trouble until I came across Tim Gowers' blog post on How to use Zorn's lemma.
Its main thesis:

If you are building a mathematical object in stages and find that (i) you have not finished even after infinitely many stages, and (ii) there seems to be nothing to stop you continuing to build, then Zorn’s lemma may well be able to help you.

He shows how to use the Lemma in a number of cases where my intuitive approach would have been something like "ah, but we can construct the thing by transfinite induction ... let's find a sufficiently large ordinal to induct over (work, work, work) ... and then fix a choice function such that we can make choices at each step along the way (work, work, work) ... and if the thing is still not made when we reach the top of our chosen ordinal, it would be (work, work) a contradiction".
Compared to that, Zorn's Lemma packs a lot of boilerplate argument into a simple, tidy, reuseable tool where one just needs to specify the minimal properties of the situation for the construction to work. In particular the apparently ill-motivated condition about chains is exactly what is needed for the tedious transfinite-induction argument to keep rolling when we hit a limit ordinal.
