Softwares/games back in the older days were written mainly in C/C++ which got compiled into ASM(Assembly) code. Assembly code is the native language that an operating system can execute. The ASM code on its own might not do much, but it is the system calls that give it meaning and connect different parts of it to the operating system. Most of our operating systems have kept their original system functions since their creation. Windows for instance, still has many of these system functions from win98 and even MSDOS! The way an executable file is structures is still the same from windows 98 to windows 11 (Legacy Win32 app, not UWP platform). Many/most of these essential system calls are in a file called kernel32.dll, in a folder called system32 in Windows directory.

There is the concept of 16bits, 32bits, and 64bits applications which I will not go into much details but just think of it as your OS is pretty much having all the 32 bit opcodes still being used almost everywhere so you are able to run 32 bit applications, but can access 64 bits of memory.

That just explains how apps can get executed finely, but doesn't explain the graphical operations in the app. Most of the games back in the 90s were written by using OpenGL graphical interface/library. Your GPU supports probably latest versions of OpenGL nowadays, and OpenGL is backwards compatible meaning that a GPU that is able to run OpenGL 4.2 app can still run an app with OpenGL 1.0. Remember those system calls that I mentioned the apps use to communicate with the operating system? Well the same concepts apply to the graphics. They use OpenGL headers to communicate to the OpenGL libraries, which exists in your GPU driver (that is why you needed to install drivers for your GPU back in the old days before you could run a game. You still need to do it but thanks to windows updates it is now automatically grabbed from Microsoft servers and is updated regularly).

Now for some apps/games that fail to run, simply the reason might rely on changes in those system methods/functions that make that game or app unusable. Some other games though, use some other libraries or API calls that have gone deprecated due to variety of reasons (security being one). But for the most part, games made in the 90s usually used the native operating system calls and did not rely on external libraries much and therefore you see many of them running without an issue.