You don’t need to have 18” biceps and weigh 300 pounds to arm wrestle, unless you’re arm wrestling a person with 18” biceps who weighs 300 pounds. Then, yes, you kind of do. Because otherwise, that guy’s giant hands essentially wrap around your entire lower arm down to the elbow, and that just doesn’t work.
But except for that, it all comes down to conditioning and technique.
Conditioning:
You need to build bicep strength for pulling, forearm strength to keep your opponent from straightening your arm, and pec and shoulder strength for rotating your opponent's arm down toward the table. You also need to work your obliques and other core muscles, which contribute to the side-bend motion, as you bring your body weight to bear. Less obviously, you also need to strengthen your tendons and other connective tissue. Exercises include:
Isolation bicep curls
Hammer curls
Cable crossover
Wrist curls
Side planks
Static holds– Once you've elevated a weight, hold it in a stationary position to maintain pressure on the connective tissue supporting the hold
Light weight, high rep wrist and forearm exercises also strengthen connective tissue
Technique:
Pull your opponents arm toward you, to isolate it from his body, while keeping your own arm as close to your body as possible
Turn your wrist so your knuckles face toward you
Then continue pulling and pushing down, without turning your body.
Go Further:
Here are some great online resources where you can learn more: