As explained in our FAQ as well as Wikipedia, for most FOSS software you can download and install the binary from its repository. To do so, you will have to allow downloading and installing apk files, which is a security risk, but it is the only fastest way to get this app on your phone. Make sure you save it to internal storage and not the an external card (if you have one), and after install, revert back to disallowing apk file installations. This faster way to install (e.g., from GitHub; also called "installing from source") might make updating it more difficult.
Some users are concerned about security of FOSS apps. Those who use them believe that they are more secure, criticizing the "security by obscurity" model. It is important to note that apps in both the app stores of Google and Apple have suffered serious security issues; furthermore some apps are listed in both Google's Play Store as well as F-droid and they are often quite secure.
Most FOSS apps can also be installed from the F-Droid repository, which is an app "store" similar to those owned by Google or Apple, except that it is dedicated solely to open source software (aka OSS). You install F-Droid by downloading the F-Droid apk from the official website, and by allowing the installation of apk files (as above). To complete the install, we recommend to download to internal storage, allow installation of apk files, complete F-Droid installation, allow installation of apk files from F-Droid, then go back to preventing apk installs from the browser. Needless to say, you will encounter a number of warnings that you will have to override, so if you are unsure or concerned, this might not be your best bet. Additionally, some apps (such as Open Camera, Termux, Linphone) can be found on both F-Droid and Google Play, and although both app stores will alert you when there is a new version, you must remember to continue to update them only from the store you first installed from. You can find some Youtube videos on installing F-Droid: (landscape: Websplaining 3:33, DanielTT 6:24; shorts: Learn, But).
F-Droid may be used with its official client as well as through other clients such as NeoStore, Flicky, Droid-ify and G-droid; Aurora is a FOSS client for Google Play Store.