I could find one letter to the editor from 2011 that posits a few mechanisms by which increased blood flow caused by the fever might make autism symptoms better: Does fever relieve autistic behavior by improving brain blood flow?. It also mentions a workshop on the topic that happened in 2010.
Here is the workshop report.
One clear conclusion from the meeting is that much more research is needed to investigate the relationship between fever and autism, in particular whether the fever-related responses are a result of the temperature changes, or of a neuro-immune response related to infection. Clinical tests can examine the role of temperature by monitoring the effect of safely raising temperatures (such as in a water bath) on symptoms
There is a reply to the letter to the editor with some more thoughts related to causes.
I could find nothing newer, but this year, a grant of 900,000 USD grant was awarded to Indiana University to research that link: NIH awards Indiana University $900,000 to study link between body temperature and autism
"Like many research topics, the phenomenon isn't totally unknown, but exact mechanisms linking body temperature and autism haven't yet been organized as a principle and unpacked to see how it could work," said Alberts.
The IU study will be conducted in mice. Alberts and Harshaw will investigate the association between physiological deficits in the ability to regulate body temperature and social behaviors associated with autism using mouse models for both conditions.