Weapons in Red Dead Redemption 2 wear down gradually. They can also be worn out when you first get them. The game allows you to repair them. This page contains information on how to repair weapons and what effect does this process give.
To fix a weapon in RDR 2 you need to clean it by removing dirt or rust. This allows you to get back a weapon's stats back to its original values (damage, fire rate, reload etc.). It should be noted that cleaning weapons is an optional step. The penalty for using a worn out/dirty revolver or a carbine isn't high - their stats get lower rather slow.
You don't have to check your weapon regularly. The game informs you when a weapon needs cleaning. To repair a weapon you have to open the wheel menu and find the one you want to fix. Press the button that opens the cleaning menu - press the right analog stick (the default button).
Press the cleaning button when Arthur starts examining his weapon (Square on PS4). You need Gun Oil to clean a weapon. You can find it on corpses (try to always search dead bodies) or you can buy it at a store (General Stores and Gunsmiths). A Gun Oil should cost you less than $2.
If you don't have Gun Oil, you can also visit any of the Gunsmiths and pay for cleaning your weapon. This service shouldn't cost you a lot.