They don't. It's total bullsh*t.
It's like saying that toothpaste makes you gay. Imagine it is 1979 and a dentist writes an article about how toothpaste makes people gay. The article was determined fake and the dentist got his license taken because he was spreading misinformation. He couldn't be a dentist anymore. And now imagine that till today in 2019 some parents would still say: 'I don't brush my childs teeth because I don't want them to be gay. I don't care if they get caries or lose a tooth I just don't want them to be gay.'
That would be ridiculous right? But that is the same thing that happened with vaccines and autism.
Autism is something that you're born with. Symptons can be found at a later age but the autism has always been there.
Speaking of gay. I once read an article online (not scientific just for fun) about the correlation between autism and homosexuality. It said that people who are autistic are more likely to be gay or bisexuel than non-autistic people. I don't know if it is true but it did make we wonder this: if there are people that claim vaccines cause autism and people who claim autistic people are more likely to be gay, does that mean that vaccines make you gay now?
Perhaps Andrew Wakefield can write an article about it. ;)