After acquiring the tech, a draft icon will appear on the map. Simply tap it to gain some additional armies.
The number of armies you gain from drafting is the sum of two separate formulas: The fixed draft portion and the sliding draft portion.
The fixed portion is calculated based on your armies/sec number every time you draft. Therefore, the total armies gained from the fixed portion scales linearly with the number of times you draft as well as modifiers that increase your draft sizes.
The sliding portion is calculated based on both your armies/sec number as well as how many armies you've drafted in the past. The total armies gained from the sliding portion will approach but not exceed a percentage of the total armies gained in that level.