When you want to construct something very large, or just handle something very large, you can do it using the axiom of choice and/or the well-ordering theorem coupled with transfinite recursion and induction.
It's not very difficult, but one has to understand the structure of the ordinals in order to use the ordinals properly. In particular one has to know a bit more about well-orders in order to use transfinite induction.
If someone wants to prove that every vector space has a basis; or if every unital commutative ring has a maximal ideal; or whatever. They might not be very interested in well-orders, ordinals and transfinite recursions.
Zorn's lemma offers a way out. It offers working with partial orders, often we use subsets or some other objects that we know sufficiently well.
And of course, it has plenty of easy uses, since partial orders where every chain has an upper bound are abundant in mathematics (see two easy examples above). It is a lemma since we're not generally interested in the partial order or in finding a maximal element there. We are, however, interested in specific partial orders (the set of linearly independent sets ordered by inclusion, for example). So the term "lemma" is very fitting, since it tells you some nontrivial information about a general context that is applicable to your case.