The summary is here: https://www.judiciary.gov.uk/wp-content/uploads/2016/11/summary-r-miller-v-secretary-of-state-for-exiting-the-eu-20161103.pdf
The full transcript is here: https://www.judiciary.gov.uk/judgments/r-miller-v-secretary-of-state-for-exiting-the-european-union/
I will try in a sentence to sum up the reasons behind the high courts judgment:
Parliament voted to join the EU, so parliament must vote to leave the EU.
The high court is not seeking to thwart the decision of the June 23 referendum, which I for one have made clear I completely accept, as does the court; it is merely saying that we enact this decision through parliament, not by the executive power of the PM. The government is appealing this as it believes that it does not need to go back through parliament and can get on with exit via a simple decision of the PM.
On Monday this discreet matter of the route of the exit will be decided by the Supreme Court. I utterly reject criticisms of the Supreme Court and high court justices, as orchestrated by the papers. But the key point of today's blog is to make the point that it is a slippery slope to attack our rule of law. Having been a lawyer for 20 years I am entirely satisfied that the decision is to be tried on its merits. The reality is that those seeking to exit the eu fought long and hard to ensure parliamentary democracy. And the judges agree. We will be leaving the eu. It is just a discussion of the means and way we do it that is being decided. This means of exit matters - namely by prime minister alone or by an act of parliament.
If you want to watch the case as it unfolds you can watch it live here: https://www.supremecourt.uk/live/court-01.html