I've had this in the past (and still do occasionally), and I put it down to three possible issues, fitness, food and water.
If, as by your own admission, you're pushing yourself particularly hard during a set, then you may find that you work harder than your body's ability to supply oxygen to the muscles, which would result in you gasping for air and feeling dizzy if you can't get enough oxygen into your system (yes, this is a very over-simplified version of events, but it boils down to you working harder than your body can handle = dizziness).
When you train and push yourself, your body pulls blood into the muscles to fuel them, pulling it from places like your stomach, where it is normally used for digestion. If your stomach / guts / intestines can't digest the food in it, wants to get rid of it so it doesn't sit and rot, and does so by trying to get rid of the food in the most expedient manner (pretty much the direct opposite to how it got in there in the first place).
Having said that, there is still food in your stomach that's trying to be digested, so your body is probably trying to keep some blood there for digestion, causing less to be available to fuel your muscles, which is why you're getting the nausea / dizziness.
(Again, over simplified explanation of body stuff, but you should get the general gist).
You're not drinking enough / your salts / electrolytes are out of whack. I'm going to admit, I haven't read into the physiology of this, but I know from personal experience that going into a training session a little dehydrated can really screw you up. Make sure you've drunk enough beforehand and during.