$10B divided by 218 miles is approximately $45M per mile. That's not pulling something out of an ass, it's just math.
As for how they came up with that - I would assume very costly studies involving consulting experts in rail, construction, land right-of-way negotiations, transportation permitting, and numerous other expertises involved in the process. The private developer behind this has done rail before, so in theory they know how to price these things out.
It's new technology that hasn't been built in the US. There are significant unknowns and new scenarios that will drive up price.
It's $45M per mile. That's less than 1/2 the cost of urban light rail, and less than 1/4 the cost of the high speed rail being built in the Central Valley. If they can do it anywhere close to that price it will be amazingly cheap and cost competitive with flights.
The article literally says they "must pass through fossil fuels". That's not bypassing fossil fuels to jump direct to renewals. I have no idea why you couldn't follow the comparison to India's strategy for telecom buildout.
... though his invention worked superbly -- his theory was a crock of sewage from beginning to end. -- Vernor Vinge, "The Peace War"