As we all know, Aizawa's Quirk can temporarily disable his opponent's Quirks when they're within his eyeshot. While he can't physically remove mutant type Quirks, he can limit their function such Ojiro being unable to move his tail. Then the question becomes, how? What are the scientific technicalities of the Eraser Quirk? First, I'll address the most popular theory and then explain my own later on.

Popular Theory:

Due to his eyes glowing red, it's believed Aizawa's Quirk involves the Light spectrum. According to this theory, he projects rays outside visible light from his pupils, so the reader or characters are never able to see them. When those rays enter someone else's eyes, it sends a false message to their brain to numb the part responsible for activating their Quirk.

With the specifics presented by the light theory; the target making eye contact with Aizawa must be a requirement to be affected by his eraser quirk. This is due to Aizawa's supposed rays being blocked by his eyelids whenever he blinks. If his rays can't get past eyelids, it would be impossible for them to get past human bone/skin and implant a false message directly into the nerve system. Only eyes could be affected by Aizawa's rays, as the only organ that detects light.

Despite Aizawa's rays not being visible light, they still must be fairly close on the light spectrum. They're unable to be something stronger such as Gamma Rays or Broadcast bands. This is due to Aizawa's eyelids being able to block his powers when he blinks. Eyelids are a narrow barrier made from skin, so powerful rays such as Broadcast bands wouldn't be stopped by them. Meaning Aizawa's rays would either be ultraviolet or infrared, so they can't be seen with the Human eye however will still be blocked by thin barriers like eyelids.

However. There are several issues with the light theory.

1) Aizawa was shown to erase Shigaraki's decay quirk before he almost killed Tsuyu. According to the light theory, this should be impossible as Shigaraki's back is turned and the supposed rays wouldn't reach his eyes.

2) It doesn't explain how Aizawa's quirk makes his hair rise in the air. Therefore, it's usually dismissed as an unexplained side effect.

My theory:

I find the statistical likelihood of a quirk specially designed to erase other quirks unbelievable - As Quirks are a biological phenomena from random chance. There's a reason so many people have useless Quirks such as pulling your eyes out of your sockets (Real One). Natural selection hasn't had time to settle in yet as they've only existed for roughly a century. Quirks aren't sentient or aware of each other's existence, so its borderline impossible one would develop to counteract its brethren so soon after the rise of Quirks.

That's why I believe Aizawa's ability to erase other's superpowers is not a Quirk in of itself, but simply a super move. Specifically, I believe his Quirk: is a small-scale form of telepathic levitation that can solely affect living creatures, including DNA.

The activation of the Quirk is dependent on Aizawa's eyes and its strength weakens the further away an object is. It's like a heavily washed-down version of Inko Midoriya's Quirk.

The reason I especially say DNA, is due to this quote from Season 4, spoken by Aizawa himself: "What we call a Quirk is an extraordinary addition to an ordinary human body. Those additions are collectively referred to as Quirk Genes. I can shield those genes and temporality block their expression but I don't actually damage them."

To put into context how weak I believe Aizaka's levitation is; one of the select organic materials he's strong enough to float are hair strands - Which is why his hair floats whenever he erases someone's Quirk. This is an involuntary action, as Aizawa has never bothered to gain conscious control over this aspect of his Quirk. But, who could blame him? As a young child experimenting with his Quirk, literally all he could is make extremely light-weight organic materials levitate near his eyes.

If Aizawa hadn't discovered a unique/precise way to use his quirk, he would have deemed it useless and never tried to be a Hero. Due to his Quirk being so small scale, it would be a waste of time to train to lift heavier objects, as at most, he might upgrade from a hair strand to a mouse. Instead, he shifted focus to the cellular level. Cells are so minuscule, his limitation of his telepathic powers weakening over distance hardly mattered. Even Aizawa's extremely weak Quirk was capable of levitating/moving cells from several metres away.

For someone to be affected by his quirk, it's as simple as him seeing apart of their body, no specifics required. This suggests a "honing in" aspect to his quirk; so it requires minimal mental strain on Azawa's part. He simply needs an idea of what he wants to be targeted and if it's in his eyeshot, his quirk will anatomically try to levitate it. Unlike Momo needs a clear mental picture to control her creation quirk, Aizawa simply needs to know his microscopic targets are in his general area and his quirk will naturally hone in on for him.

So how could the manipulation/levitation of cells inside the human body be used in Hero combat? As decisions need to be made within seconds during life-and-death situations, Aizawa doesn't have the luxury of analysing each individual's body for specific weaknesses. He needs a method that can be used on as many people as possible. Blood clotting is a possibility, but blood is even heavier than hair strands and can cause unnecessary health issues for villains.

However, there's may multiple possibilities to interrupt quirks by manipulation on a cellular level. One way, is Aizawa "hones in" on the electrical messages in the human nerve system, those responsible activating quirks. He could have researched the differences between human nerve messages and trained his quirk to "hone in" on specific messages. If the quirk activation message between the brain and muscles is interrupted, say levitation messing with the electrical inputs, it could potentially "erase" someone's quirk.

A second option is DNA. Due to Quirks only existing from 3-4 generations, they haven't had time to fully integrate into Human DNA, so they can be tracked down to a select few genes. If Aizawa "honed in" with his levitation to temporality disjoin those genes from his target's DNA strands - He could potentially stop his target using their Quirks.

Regardless of which option you consider more plausible, either interrupting nerve messages or DNA, the result is still the same. Aizawa uses his small-scale levitation to manipulate the cells in the Human body in such a way, it temporarily stops them using their quirks. This feat would obviously be difficult to train, however if there's one thing Aizawa is good at, is relentlessly pushing himself. Vigilantes showed that as a teenager, he was willing to stay after school, well into the night, to train by himself. If he had this determination as a high school student, he could of had it as a middle schooler or even younger. He had the drive necessary to train his seemingly useless quirk until it had utility in the UA entrance exams.

It would make sense if Aizawa doesn't want to reveal to people that his quirk is technically a small-scale levitation. As he would have worked hard to develop a useful super move from a useless quirk, he'd rather people assume erasing Quirk is his sole power. Publicly exposure of how useless his base power is could somewhat affect his social life, which as a loner, he wouldn't want to deal with. He could also lose a tactical advantage in hero combat, as his stealth relies on a fear factor. If Villains discovered his base power was so weak, they might not fear him as much.

It could explain why in Vigilantes, he had an inferiority complex when it came to his quirk as a High School student. Because in truth, his quirk was actually inferior to his peers, at least in the past before he developed his erasing super move. It would take him serval years to fully accept his trained quirk was now powerful, when for most of his childhood, it wasn't. It may explain why Aizawa was lagging in his combat skills during his first two years in UA school. While preparing for the UA entrance exams in Middle School, Aizawa might have spent his training perfecting his erasing super move, leaving him no time for general combat.

In conclusion, that's how I believe Aizawa's Quirk works, small-scale levitation, that he developed an incredibly specialised super-move for, before entering High School.

Bonus:

Even if this "bonus" can be discredited, it still wouldn't affect the main theory. Which is an explanation for Aizama's support item strange ability to levitate, eerily similar to his hair. Despite it being a support item, its ability to float and being seemingly mentally controlled at times, is too advanced compared to other technology within the MHA world. To fly, they still have to use mechanical jetpacks the size of a backpack. If the scarf could float via technology, shouldn't it have a device attached to it? Yet it doesn't. It's a plain piece of material.

In the Manga, Hitoshi Shinso was shown to have the same support item as Aizawa. Though one key difference is that Shinso can only use it by physically using his arms, he is never able to make the scarf float. If Aizawa has a levitation quirk, this could explain why. The scarf is made from carbon fibres and a special metal alloy - Those materials are notoriously light-weight. So even his weak quirk, if pushed, could levitate this light-weight scarf, similar to his hair.

Just like Mirio, Aizama could of had his scarf costumed made, by having his hair woven in. Therefore, the DNA inside it would allow him to control it with his quirk. This would give Aizawa the needed boost to keep up with other students attempting to become heroes, allowing him to join UA School.