The short answer to your question is that it takes time to mount an antibody response.

Vaccines do one big thing: they "teach" the adaptive immune system how to produce antibodies for the virus, and in the process the body produces a ton of antibodies, which fade over time. The antibodies fade quickly-ish (months), the "learning how to make antibodies" fades much more slowly, if at all (decades).

1) You get symptoms if you don't have enough antibodies in the right place when the infection comes in.
2) You avoid severe illness if your body can produces enough new antibodies (if it doesn't have enough) fast enough before the virus infects the really vulnerable parts of you.

People who have symptomatic breakthrough infections probably didn't have enough antibodies left in their noses to counter the initial infection, but are able to very rapidly mount a new antibody response so they don't get sicker. The initial 100,000x overkill is full-body, and fades over time. Months later, a vaccinated person might not have enough antibodies floating around in their nose to completely neutralize an airborne bubble with 10,000 virus on it landing inside.

As someone else implied elsewhere, the immune system is super complex. I'm going to attempt an ELI15 here. Apologies in advance for the wall of text.

== A Story ==

There are two parts to the immune response, called "innate" and "adaptive".

The innate response are first responders. They kick in as soon as they notice the "wrong" thing floating around in your nose/airway/lung cells. This response produces the early symptoms associated with respiratory infections: inflammation, sneezes, coughs, sore throats, runny noses, fevers, etc. It's trying to get as many of the body's white blood cells (immune cells) into the area as possible to clear out this virus that is simultaneously infecting cells and reproducing like crazy: doubling every 20 minutes; by the time the innate system kicks in, your hijacked airway cells have probably produced millions of new virus already. The more virus, the stronger the innate response: more immune cells, more killing infected cells, more clearing of cellular debris, etc. Your nose is runny, your throat is inflamed, etc. -- it's a huge battle, with millions of your cells fighting billions of virus. The best they can do is try to stem the tide, and they very likely won't win on their own.

Fortunately for you, during all this, some of your cells take parts of the virus, and present it to the cells responsible for the adaptive immune response. These are the scientists sitting in the back lab figuring out how to produce "antibodies" -- these are the red shells in MarioKart. They are designed specifically to latch on to and inactivate the invading virus, and make it super easy for the rest of the immune cells in your body to recognize the virus as foreign and destroy it. Designing these antibodies takes some time, unfortunately, and meanwhile there's a huge battle going on for your airways. Antibodies are HARD to design: you want something that attacks ONLY the virus, and not, say, also your pancreas by accident, because then you get Type 1 diabetes, and, well, that's not so great.

There is a lot of time pressure, because the virus is winning, and if the back lab scientists have never seen anything like this virus before, designing the antibodies can take days, maybe even a week or more, and then producing them in mass quantities takes days longer. The virus could overwhelm the body in the meantime.

BUT! But, if your adaptive immune system's scientists have seen the virus before, say because you were vaccinated -- a process in which your immune system was presented with a bunch of parts of virus that were taken to the back lab and the scientists designed antibodies for it, WITHOUT the time pressure of an actual infection -- then they basically have blueprints ready to go. (Less time spent coming up with plans means less time for the virus to overwhelm you.)

Now, regardless of where your adaptive immunity learned to make antibodies, when your body is ready, it goes to TOWN: we are talking trillions of antibodies produced in days, they literally flood your whole body. At peak there could be hundreds of thousands of antibodies for every virus particle. At this point, unless the virus figures out a way to hide from antibodies, it has more or less no chance. It's over. You've won. Your body will still need to clear the battlefield, so you may still have symptoms for a bit while this happens.

For vaccinated folks, there may even still be antibodies floating around from the vaccine, making everything much easier. There may be so many antibodies floating around that the virus doesn't even stand a chance -- it gets neutralized really fast (and transmission is blocked); there may just be a few antibodies around, but the adaptive response kicks in fast (mild symptomatic illness) and wipes the floor a few days later.

I hope this helps paint the big picture (with timelines) of what happens when a virus lands in your nose. I'm sure the HN crowd can imagine all sorts of edge cases here, and when hundreds of millions of people are infected with a single virus, basically any edge case with probability greater than 1e-6 is going to happen >100 times.