Both male and female African dwarf frogs reach sexual maturity at about nine month of age.
Female ADFs are a bit larger and plumper than males, especially when they're gravid (carrying eggs) or "eggy," like the female frog in this picture. They also have a cloaca at their rear ends, which is visible as a small bump between their hind legs. They use this opening for both laying eggs and passing wastes.
Mature male ADFs have subdermal glands that are barely visible as white or pinkish bumps on the backs of each of their front legs, close to what would be their armpits if they were human. These glands have something to do with the mating process, but we're not sure exactly what that role is.
Males also "sing" when they're in the mood for love. Their singing can often be faintly heard outside the tank and sounds like humming or buzzing.
African dwarf frogs mate using a method called amplexus, or amplexing. The male frog mounts the female frog from the top and rear, embracing her by placing his arms around her waist, behind her arms. The female then starts swimming around rapidly and doing all sorts of maneuvers in the water, taking the male along for a wild ride, and laying her eggs near the surface of the water while the male fertilizes them externally.
Amplexus goes on until the female has released all of her eggs. It can last anywhere between 30 minutes and several hours, occurring in short bursts separated by "resting" periods spent at the bottom of the tank. Most likely she is moving her eggs inside her body toward her cloaca during the resting periods.
The male will continue holding on to her the whole time. When the male is embracing the female, they are said to be "amplexing", "in amplex", or "in amplexus."
When the female has expelled all of her eggs, she will go limp and may actually appear to be dead. Whether this is from exhaustion or as a deliberate signal to the male that they're finished is unknown. I suspect it's a little of each.
Whatever the case, when the female goes limp, the male will release her from his embrace, swim away, and resume his normal activities. The female will do the same very shortly thereafter. The eggs will hatch somewhere between two and seven days later.
Raising tadpoles is not a beginner-level skill and is beyond the scope of this site. But if your frogs mate and you want to give raising tadpoles a try, I suggest you visit this page to learn how. It's a lot of work, but it's not very hard work.