This is one of the reasons why I hope someone will make a guide. (There's also a reward for anyone who does make a guide for this game!) :D
What does the monster "Egg" do?
It hatches! Notice that when you picked it up, it said "protect me!"
That's just what it wants you to do! Check the egg's stat screen. You'll see that it has 1 out of a little over hundred HP. When you get it to the max HP, the egg will hatch. But you can't just go to the inn to raise its HP. You have to keep watch over your egg in the "heat" of battle until it is born. :)
Summon the egg so it's on your team. Select the --guard-- option and select the egg to shield it from attacks. If an enemy targets the egg, you can have a member of your team step in and take the attack. The baby in the egg will sense that it was protected. It will "feel the pain" of whoever protected it, and its HP will be boosted. (Keep in mind, just selecting the guard option isn't enough! It has to actually be targeted by the enemy!)
Keep it safe from harm long enough until its HP is maxed out, and a newly born beast will crack its way out of the egg. (The ally is not random.) It will then treat you as its master. (Mother!?) There are several eggs hidden throughout the game!
P.S. Here are some personal tips of mine when raising eggs.
1. Put your egg at the front of the lineup so that it takes more hits. 2. Wait until the moon is full (demons won't talk to you during battle this way) 3. Enter a cave. The lunar phases won't move while you're in a cave! 4. Keep your team as small as possible! More allies mean more choices for the enemy to attack other than your egg!
Since it's a full moon, you can concentrate on guarding the egg easily without having to worry about the beasts trying to talk to you and ending the battle. After all, you want them to try to aim for the egg, instead!
Note: (I think the ideal size for egg hatching is three members including the egg, because you may need to end the battle to heal or something. If that's the case, you can use one member to attack, and the other to guard. After all, if it's just Yuri and the egg, who will protect it when he attacks the enemies?)
The egg gains HP by doubling the damage that the one who protected it received.
So if Larsa takes 1 damage from an enemy attack, the egg will gain 2 HP. If Larsa takes 2 damage, the egg will gain 4 HP. That means you should --unequip your defensive gear-- so the enemies will hit you harder and the egg will hatch faster!