First time posting here so I hope this is up to standard, but I'm a Skin, know Skins and have touched upon the topic in essays at Uni. This is from a UK perspective.
Skinheads originated out of the Mod scene of the 1960s. Mods would wear fairly expensive, middle class fashions that were not always attainable for the working classes. So working class Mods started to dress in a much more "stripped down" style, work boots, straight leg jeans, shirts and the iconic braces (suspenders).
The shaved head appeared due to a few reasons, firstly for safety, long Mod style hair could be a hazard in a factory or in street fights and secondly it was a way to rebel and differentiate themselves from other youth tribes such as Mods, Rockers and the early Hippy scene.
As for the racism, that's complicated for many reasons. But the Skinhead subculture grew out of working class unity between the traditional white working class and the newly arrived Caribbean immigrants, they shared music, mostly notably Ska and early Reggae (during the early 70's a sub-genre of Reggae was even dubbed "Skinhead Reggae" in order to appeal to a British audience), they also shared styles of fashion, with many working class youths copying the so called "Rudeboy" fashion popular amongst the new immigrants.
During the 70's and carrying on to the 90's, far-right groups were actively recruiting white working class youths at football matches and clubs, if I remember correctly the National Front actually held free club nights for members in order to draw in more recruits. Many Skinheads, along with many others fell for and believed what these groups were saying. The early 80's saw a number of incidents that solidified the notion that all Skinheads are racists. Perhaps the most famous was the debacle over the "Strength Thru Oi!" Compilation, the cover of said album had an image of the openly racist Nicky Crane, a poster child of the far right, that image and the title, being a play on words of a Nazi slogan didn't help matters. This period also saw an increase in violence at gigs, Sham 69's farewell gig was marred with violence perpetrated by the far-right.
I should point out that, yes, there were far-right Oi bands but also left wing ones too, Angelic Upstarts and The Oppressed are two examples from the UK and The Templars from the States. Also SHARP Skinheads Against Racial Prejudice) Skinheads are still a thing but not as coordinated as they once were, it's much more an individual classification.
Anecdotally, if a subculture or group has a reputation for being violent or racist then it will attract violent people and racists. I've lost count of how many people I've met who call themselves Skinheads because they're racist and feel like the wider Skinhead community will condone their actions. But I think this attitude is changing, films like This is England and newer Skinhead bands like Gimpfist are doing a lot to rehabilitate the image.
I hope this all makes sense.
Sources: Spirit of 69: A Skinhead Bible by George Marshall
I could follow up with a similar question: My understanding is Skinheads were not originally necessarily neo-Nazis. When and why did Skinheads start to become associated with them?