- Most old products that passed mainstream support might experience significant changes on how they are supported by Microsoft before end of life. In your case, VS 2013 passed that day on April 9, 2019 https://learn.microsoft.com/en-us/lifecycle/products/visual-studio-2013
- The initial user experience of VS 2013 is that upon installing you get a month of free trial and you need to request a product key free of charge by registering via a Microsoft link so as to continue using it after that trial period. For some unknown reason (maybe your network connection, ISP, or on Microsoft side) your machine does not seem to be able to connect to that registration page.
Using such an old product has its own risks, as almost all technologies/frameworks it supports are also out of mainstream. The code/app you write there might not be able to run very well on newer platforms.
Since VS2015, Microsoft decided to end the feature limited Express edition and ship a much more powerful Community edition for individuals and small business owners,
https://visualstudio.microsoft.com/vs/community/
and they are much easier to activate if you try VS2022.