Question: Banach's Fixed Point Theorem proves uniqueness of fixed points. Suppose the contraction map $g$ had two fixed points $x$ and $y$. Then,

$$|x - y| = |g(x) - g(y)| \leq L|x - y|,$$

which implies $(1 - L)|x - y| \leq 0$. But since $(1 - L) > 0$, we can deduce $x = y$.

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* I get that the first equality follows from the definition of a fixed point
* Second equality follows from definition of contraction map
* How does it imply $(1-L)|x - y| \leq 0$, and how does $(1 - L) > 0$ lead us to $x = y$?

Answer: 
In my opinion, you should have a look at Chapter 2 and 3 of Munkres.

Another classic textbook is General Topology by J. L. Kelley, but it is about 50 years older than Munkres. As mathematician, Barry Simon, once wrote (in 1980) "For the reader who wishes to delve further into the realm of general point set topology, we recommend Kelley's General Topology most enthusiastically. The best way to read the book is to do all the problems; it is time consuming but well worth the effort if the reader can afford the time."

I also firmly believe in the advantages of having an assigned textbook for upper division mathematics. You should see the references/bibliography of your lecturer's notes (if it exists).
I have two recommendations: 

* *Introduction to Topology: Pure and Applied,* by Colin Adams and Robert Franzosa. (This has lots of examples. It covers all the material one should see in a one-semester course in introductory topology. It also includes lots of interesting applications.)
* *Introduction to Topology,* by Crump Baker. (This is a *very* underrated introductory book, in my opinion. It deserves more attention than it has received.)

The book by Munkres is great. However, if a person (like yourself) has been struggling with proofs, then the book by Munkres might be better for future reading.