Yes, phones can emulate DS games. This is done through a process called emulation, which allows one system to imitate another. Emulation has been around for quite some time, and has been used to allow computers to run software designed for other platforms. In the case of DS emulation on phones, this means that phone users can download and run DS game ROMs on their devices.
There are a few different ways to go about doing this. One option is to use an app like DraStic, which is available for both Android and iOS devices. This app will require you to purchase it in order to use it, but it does offer a high degree of compatibility with most DS games out there. Another option is using a web-based emulator like DeSmuME Online. This emulator doesn't require any installation or setup, and should work with most modern browsers. However, its compatibility may be somewhat lower than that of an app like DraStic.
Either way, if you're looking to play DS games on your phone, emulation is the way to go!