In America, we tend to think bigger is better: big cars, bigs drinks, big everything... But, for me when it comes to cameras, big is a huge pain! I like to travel as light as possible and a huge DSLR made that virtually impossible. Weight is an important factor when deciding whether to take your camera with you or leave it behind. When you have a DSLR you begin to plan your life around its size: "which bag am I going to need/ which lens should I grab/ do I really feel like carrying all this stuff?" In the end, if I was going out casually (on most occasions) my camera would be left behind. The major problem with that is: no camera no photos!
The second and perhaps most important reason for me abandoning the DSLR world and embracing small mirrorless cameras is the fact that DSLRs are simply intimidating. When it comes to street photography, you want to be as inconspicuous as possible. There's just something about DSLRs that scream: "hey, look at me, I'm a photographer!" People immediately take notice of you. Once that happens, they instantly stop what they're doing. A multitude of questions seem to be etched upon their faces: "what is this person doing/ why are they taking my picture/ where is this photo going"...moment ruined!
When you have a small camera, people almost pose for you. They're encouraged not only to be in the photo, but be part of the photo; you're less likely to be viewed as some random person potentially exploiting their image. Since switching to a small mirrorless system people, have even asked me to take their photos while in the street! This never happend when I was lugging around a huge tank of a camera. I find that the relationship has transformed from "the photographer and the subject" to a much more personable and approachable one.
The same holds true for standard portraiture. When shooting "normal" people (non models), having a big camera and big lens in their face isn't natural. It takes them time to get comfortable. I have found that with my mirrorless system, there's just less machine in the way making it easier to connect.
Despite the reputation that DSLRs "look more professional," I have found that my small camera has broken down a barrier of sorts, opening up new possibilities for me that just didn't happen before.