That's not true. It can run client-side on a WebAssembly-based .NET runtime or server-side in ASP.NET core.
I find MS has generally been taking a Manhattan-project style approach these days, trying everything at once and keeping what sticks.
https://docs.microsoft.com/en-us/aspnet/core/blazor/hosting-...
I don't know if thats a disconnect in marketing, or me not paying attention, but like I said I was really confused.
But the reason they're pushing server-side Blazor is because loading client-side Blazor just takes too long in production deployments. Right now, you have to wait several seconds before the page loads, which is unacceptable for most users. I feel like server-side has to be the future, though, and they'll address the speed issue soon enough.
The WASM side of it is really interesting for me as a Xamarin proponent, because it has the potential of true single shared-codebase between mobile and web. I'm not sure why I would use Blazor right now though.