It isn't something you feel, per se. It is something others make you aware of. Once you have the self-awareness that you are smart, everyone deals with it differently based on life circumstance. Typically you find out fairly young, which means you are still very impressionable. For me, I was told I was "highly gifted" in the first grade. This meant my IQ exceeded state maximums for "gifted" and I tested beyond typical magnet programs' capacity to deal with me. Hooray. So they put me in a special educational program in the 2nd grade, and I got to be an anomaly at every school I went to until college.
How did it feel??? I felt like an outcast. By the third grade I was learning sixth grade math, but others in my class were learning 9th grade math. I was learning complicated grammar rules but others in my class were learning foreign languages. Meanwhile we were all ostracized by the rest of the school as freaks. So... how did it feel? Lonely. I was somewhere between "freakish genius" and "normal", which meant I didn't fit in very well anywhere. It took a long time to come to peace with what the educational system accidentally did to me. If my parents hadn't forced me to do socially diverse things, I'd probably be a shut-in today as a result of those early scholastic years.
As I grew up and found my own identity outside of intelligence, the fact that I was smart became largely irrelevant. It turns out that wisdom eventually overtakes intelligence and a great swath of humanity starts to equalize the older we get. How does being smart feel to me now when I am in my 40s? Well, I'm better at remembering movie trivia than a lot of people, and I can do neat tricks like take a six week online course on Quickbooks in 3 days and absorb enough knowledge to convince a CPA that I have bookkeeping experience. So it's good for job interviews. But aside from that initial edge, I think I probably feel the same as anyone else. I have a slight advantage in that I can adapt quickly to new situations, but I'm not sure if that is a function of intelligence or just a skill I also happen to possess. Also... I overanalyze... everything. Sometimes I can get locked in a feedback loop when trying to make important decisions. My wife (who is smarter than me... Gah!) and I sometimes analyze a quarrel to such a degree after it is over, that we get into a fight about the fight! I have been known to flip a coin to break my analysis paralysis. Being smart can feel like a liability sometimes, heh.
And on that note, I'd also say that based on many of my smart friends, smart pets, and my own experience... a blessing of intelligence almost always seems to be accompanied by a curse of neuroses. Smart folk (and animals) are... delicate in the brain pan.
I often wonder if I would feel differently if I had gone to a normal school and breezed through it without feeling challenged... what sort of person would I be today? Better? Worse? Certainly it would be different. Or perhaps... I'm over analyzing.