Egg fragments are an item that can be used at the Stables and pieced together in a 3x3 table, in order to hatch a Monstie.
Any eggs hatched with egg fragments will not have any extra stats compared to an egg from a den, and they won't have any genes that the hatched species has by default.
An egg fragment can have any combination of pattern, color, and position, and all of those combinations are equally likely. Also, you can only hold 100 egg fragments at a time (excluding any already placed in the egg table), so keep that in mind!

These can be entirely overlooked for the most part, although creating an egg from fragments is required for a Rider Card Achievement, if you're aiming to 100% the game.

Egg fragments can be found in one of two ways:
The first is bone gathering spots. Darker colored bone piles are rare gathering spots, and tend to have a much better chance at yielding egg fragments.
Common gathering spots and rare gathering spots don't have identical item pools, even in the same area as each other. Some items can only be found in common spots, and vice versa.
The items you can find in a gathering point vary by area, so some places won't have any egg fragments at all.
This also means that even if you can find egg fragments from the rare gathering spots in an area, the common gathering spots there might not have any whatsoever.

The other way of getting egg fragments is battling another player. If you win, you'll receive 3 egg fragments, each of which will match the genus of the opponent's Monsties.

If you have someone to play with, trading wins with each other is a much more efficient way of getting egg fragments compared to gathering them! You can speed up the process by fleeing at the start of each battle. And if you're looking for a specific egg fragment pattern, then have your opponent bring Monsties that match it. There isn't any way to influence the colors or position of the egg fragments, though.

All egg fragments have a pattern that matches a genus of Monstie. This determines the species of the hatched Monstie.
When you hatch an egg with multiple patterns in it, the game will pick one of the nine placed fragments, and then use this to decide the genus. After that, it picks a random species from said genus, and that's what hatches.
For instance, if an egg has 8 Amphib fragments and 1 Fly Wyv fragment, then it will have an 8/9 chance to hatch a Zamtrios (the only Amphibian monstie in the game).
There aren't any species/subspecies that are more likely to be chosen than others, though. All species are equally likely to hatch. An egg made of all Fly Wyv fragments is just as likely to hatch a Silver Rathalos as it is a Khezu.

Egg fragments can come in red, blue, and green, which correlate to ATK, DEF, and HP.
If you line up three egg fragments of the same color, you get a bingo! This will give you one hatch bonus in whichever stat that color bingo corresponds to.

Just like with hatching eggs from dens, a hatch bonus in ATK is 1 point, DEF is 2 points, and HP is 3 points. The maximum amount of hatch bonuses is 8, which is also the exact amount of bingos possible in the egg fragment table.
In order to get a Monstie with all 8 hatch bonuses, you need to have all nine egg fragments be the exact same color as each other.

When finding egg fragments or receiving them after a battle, there's a chance you'll wind up with a rare one. The game never tells you these exist, or what they do, but they only really have one purpose: hatching Monsties with extra open gene slots.

When you hatch an egg from only the usual egg fragments, the hatched Monstie will always have the default/lowest amount of gene slots that species has. But if you use a rare egg fragment, then the hatched Monstie will have an extra gene slot comapred to what it normally would have.
Each rare egg fragment contributes one additional open gene slot.

This also means it's pretty overkill to make an entire egg out of rare egg fragments, since you'll be using up several of them to no effect. The lowest amount of open slots something like Aptonoth can have is 4, which means using more than 5 gold fragments on any Monstie genus is also a waste.

There also isn't any reason to try to farm these, because it's almost always much easier to just clear some floors in the Tower of Illusion to get a Stim item for the genus of Monstie you're trying to hatch.