First, you need a mercenary egg.
There are three available ways to obtain a mercenary egg.
1)Normal mercenary eggs are items you receive from the death of wild Digimon you see in the maps. You must kill repeatedly any form (rookie, champion, ultra or mega) of the Digimon you want to obtain until they drop a shiny big egg. This can be fast but also very long.
Those normal eggs, when you try to hatch them, are easy to break. You must at least collect 5 to 8 in order to succeed. If you want a bigger and stronger digimon, this may raise the necessary amount to 50. Max. size can be achieved after more than hundreds of eggs.
2)The easiest way is to buy an egg at the Cash Shop (for real money: 7dollars an egg; with free charge offers: answer to surveys or sign in in some websites). You get an unbreakable egg. But you do not choose the level (size) of the egg, it is randomly decided after the purchase. (3/5: inferior size and stats, 4/5: bigger Digimon with more ATtack/DEfense/Health Points, 5/5:biggest Digimon with a bit more AT/DE/HP than 4/5).
Those eggs are also available for game money, a lot of players buying them in CashShop only to sell them. (in this case, you might choose the egg level, of course the rarity of 5/5 make them the most expensive while 3/5 are very cheap)
3)Last method: Joymax gives away a lot of reinforced eggs (those from Cash Shop) to players each month. You get one when you start a new character, one when the character reaches a certain level. There is also one or two ways to get an egg with the monthly events. Plus the facebooks games (quizz, screenshots contests...) where you can sometime win an egg.
Second thing: you need datas, also dropped from Digimon of the right type. You must place the egg in the incubator machine and input datas. Each input costs a certain amount of datas (see details on the egg's infos). Each input can randomly fail or succeed: a failure can break the egg (only normal eggs can fail). You must AT LEAST succeed three times to hatch the Digimon, but you can try again after that to get a bigger and stronger Digimon.
Third thing: when the egg reaches the level you wish,you need to have a free Digimon space with you (you must have a maximum of two Digimon at your side). Click on the "hatch" button: you'll see a dedicated screen showing its evolution then obtain your new Digimon and its achievement.