According to the City Council and contractor (Barhale), the closure is so that the remaining refurbishment works can be carried out in a COVID-safe manner. The suddenness of the closure apparently follows the Scottish Government's 11 June announcement, permitting the resumption of such work (originally planned for completion Spring 2020).
There is no access to Water of Leith steps, and the alternative routes are via Coltbridge Terrace / Roseburn Terrace (or parts of Quiet Route 8).
Correspondents have questioned: whether full closure is needed in order to permit 2m working space; whether the closure needs to be now; whether sufficiently advanced, visible, or widespread notice (let alone consultation) was attempted.
For more on the refurbishment, use the Search box on this page. Try: Coltbridge Viaduct.
And for more in general about this viaduct see: https://www.railscot.co.uk/locations/C/Coltbridge_Viaduct/
The Viaduct is also described in our Self-Guided Walking Tour of Coltbridge and Roseburn.