Outside of extraordinarily stressful situations I only get anxiety when I have an aura (a sort of collection of warning signs that tell me I am about to have a seizure) so my anxiety might be slightly different than the norm, but here's what it feels like for me.
It starts in my chest, at first almost like a catch in my breath or a skip in my heartbeat. Then a sickly, hollow feeling begins to creep over my chest. You know the feeling of pure happieness, like an expanding balloon of joy in your chest that is about to burst? It is the exact opposite, instead of the warm fuzzy baloon it feels like there is a sucking black hole threatening to swallow my heart. This is not just an emotional feeling either, it is physically uncomfortable and makes me feel like I need to have my hands on chest or the hole will consume me. My heart starts to race and beat almost irregularly, which of course convinces me that I am about to die despite having this happen on a weekly basis. The pounding of my heart in the grip of the black hole cause my ears to either ring loudly or to hyper focus on noises around me, I prefer the ringing because the focusing makes my head spin from sensory overload and sometimes leads to minor hallucinations. (though that part definitely has more to do with the oncoming seizure)
After that it has probably been about a minute since the anxiety started and I am usually starting to get cold sweats and the hollow feeling in my chest feels like a shop vac now, this happens all the time but somehow something feels off (it's not) and I basically go into an inward downwards spiral convincing myself that I am not going to wake up from this one, something is wrong and I am about to die. Que the rapid breathing which does wonders for my already spinning head, I usually start getting really dizzy at this point and if I am now at least sitting I may faint or stumble into things. After that its really hard to tell what is anxiety and what is my brain glitching but usually the response is to lie down and curl my knees to my chest until I inevitably don't die and think about what an idiot I am for thinking I would.
On the upside the anxiety beforehand usually gives me about a minute warning to get someplace I can sit or lay down, so in a really odd way I am thankful.