The quick answer to the said question would be that Microsoft Edge cannot be directly downloaded and installed for Windows 8.1. This is because Microsoft Edge is not a standalone application unlike Microsoft Internet Explorer, and is designed solely for Windows 10.
Once again, I would like to emphasize on the term 'directly' as there is an alternate indirect way to access Microsoft Edge on other operating systems. Before your mind runs into the prospect of downloading Microsoft Edge through illegal means (by downloading a pirated copy and applying a crack to the same), I would like to clarify that Microsoft Edge can indeed be run on Windows 8.1 by legally, with the authorization of Microsoft.
Microsoft has come up with Virtual Machines for Windows 10, which is basically a Windows 10 development environment that has created by Microsoft to facilitate the development of apps for the Windows platform by developers. It provides you with a pre-configured version of Windows (the latest version, of course) along with developer tools, Software Development Kits (SDKs) and samples.
There is an evaluation version of the Virtual Machines for Windows 10 Enterprise, available at https://developer.microsoft.com/en-us/windows/downloads/virtual-machines, from which you can download the same. These Virtual Machines come with Microsoft Edge which you can use for a limited duration (90 days). Yes, it comes only for a limited duration, after which you would be required to obtain a valid license.
However, this provides you with an opportunity to experience browsing on Microsoft Edge, which Microsoft claims to more secure and faster than Microsoft Internet Explorer, so that you can make an informed decision regarding the purchase of a valid license.
Once you have downloaded Virtual Machines from the aforesaid link and opened the same, you will be required to choose the suitable Virtual Machine (Kindly choose Microsoft Edge on Windows 10 Stable or the Preview version) as well as the platform (options include: VMWare, Hyper-V, VirtualBox and Parallels). Once the installation is complete, you will be able to run Microsoft Edge on your Windows 8 and Windows 8.1 systems.
Please bear in mind that any data saved within the Virtual Machine will automatically get erased on expiry of the evaluation period. Hence, it is advisable to maintain an extra copy of the Virtual Machine image. This will ensure that you do not lose any data. Further, you shall be able to use Virtual Machines for an additional duration of 90 days.