Getting used to this can be tricky. We are very trained to think about "saving the world" whenever we see magic spells, giant monsters, epic swords, and the like. Reversing that can take a bit of effort, but when you think about it, even the greatest stories we loved as a child aren't so interesting because So-and-so killed the evil monster and saved the world, but because in the process of doing this, So-and-so learned a valuable lesson, went through important changes, and realized more of their innate potential. We all know that in real-life, no one person ever saves the whole world -- the "hero myth" is just a metaphor for that inward struggle all of us make on a daily basis to try and solve our problems, learn from our mistakes, and transcend the mundane banalities of existence.
In PvE, we get to play out the hero myth in a very direct way, but in roleplaying we focus more on the details, the reasons, and the relationships that make our heroes who they are. Roleplaying basically boils down to the following question: what problems arise when our characters start talking together, and how do they solve these problems, or fail to solve them? To illustrate, I'll give you a couple examples of how some of my characters have developed through the various relationships they've had.
When I first created my draenei hunter, I roleplayed that she was entirely new to Azeroth and could not speak a word of Common. I had a great time having her try to figure things out with body language, maps, and a handy phrasebook for a while before she gradually picked up more and more words. By the time she reached level 70, she settled into a mostly proficient but heavily accented version of Common that she just mashed together by being friendly at every opportunity. Her unique way of talking is always a great deal of fun. Incidentally, I made her pet also able to talk, through various magical enchantments, with a voice of a grumpy old man. So with this character I sometimes get to roleplay two characters at once, each one contrasting nicely with the other.
Another of my characters, a druid, had the opportunity to get married, have problems in marriage, and eventually get divorced as well. The two of them got to know each other during their early adventures together, but then eventually went separate ways as the fortunes of war (i.e. raiding) took a heavy toll on the husband's emotional well-being. Since my character was female, it gave me a special opportunity to put myself in a woman's shoes and to consider the multitude of ways a woman might think and feel in such a relationship. Of course, I would never claim that I have a perfect understanding of women from this (as if anyone does), but now when my female friends talk about some of the things in their lives, I feel more sympathetic and familiar with what they are going through. I also learned from the general mistakes that my character made in that relationship and resolved not to make them in my own life -- all without the emotional turmoil that comes from such things in real life. (In case you were wondering, I was always very clear with the "husband" that the relationship was entirely in-character, not real at all -- whenever sexual situations were about to come up, we always used to "fade to black" so that we wouldn't actually have to do any ERP.)
As you can see, roleplaying these characters involves the kinds of interactions that happen to real people all the time. The fact that these characters did quests and fought monsters was all secondary to the actual experiences they were having in relation to one another. Different roleplayers focus in more on different subjects, from romance to conflict, humor to mystery, or all sorts of other interactions. Also, these interests may change over time -- I hardly ever roleplay romantic relationships anymore, for example, simply because I'm wanting to explore new things in my characters, especially humor and cultural differences.
Whatever your style of roleplaying might be, it will invariably benefit from focusing on relationships between characters controlled by real players rather than endless fighting with computer-controlled ones. Roleplayers needn't pretend that those battles don't take place, just use them as the background rather than the main event.