Different proposals use different amounts, $1k a month happens to be in the Yang proposal (Freedom Dividend) which was funded largely from a consumption tax: https://freedom-dividend.com/
A conservative proposal ($6.5k-$10k a year, Charles Murray) might be largely funded on cuts to existing programs: https://www.wsj.com/articles/a-guaranteed-income-for-every-a...
Well I think it needs to at least be high enough that other social spending can largely be cut (except for healthcare maybe), and that seemed like a reasonable value.
Unfortunately I can't read the link you posted since I don't have an account there.
There are a bunch of UBI plans, which all vary across the political spectrum. Some are financed entirely with progressive wealth taxes, others nearly entirely with cuts to existing programs. The math usually works out decently through some trick or another, like limiting recipients to only {kids, adults} or phasing out by income.