I raise my triops in a 5 gallon (19 litres) aquarium with about 5 - 6cm of small/medium grade river gravel (from the pet store).  I use an undergravel filter which keeps the water clear and does not disturb the triops or harm the eggs in the slightest.  After a couple of generations have done their thing in the tank (about eight to twelve adults total) I break down the tank completely.

I start by siphoning/scooping all the water out of the tank until it is a few centimeters above the gravel and then I pull the plants, undergravel filter, decorations, etc. out of the tank.  I then tilt the tank at a slight angle and stir the gravel up with my hands, moving it towards the higher end.  As quickly as I can, before things can settle, I scoop out some of the rather filthy water.  I use wide mouth half-pint jars, but any shallow glass or bowl should work.  Then, I head for the sink.   This next part of the instructions takes advantage of a few properties of the eggs:

• They're lighter than the gravel but heavier than silt
• In their hydrated diapause state, they are nigh impervious to chemicals short of harsh alkalis and acids

I let each container of triops filth settle for about ten to fifteen seconds and pour off about half of the water in the container.  I then refill each container with cold tap water and let that settle for ten to fifteen seconds.  I pour half of that off and refill with cold tapwater, letting it settle for a bit, pouring half of it off, refilling, you get the idea.  What you are doing is analogous to gold panning - the heavier triops eggs keep settling to the bottom of the container after a short while but the silt remains suspended in the water to be poured down the sink.  After about four to six of these partial water exchanges the water in the container will be clear and there will be some stuff on the bottom: this is the triops eggs and small bits of gravel and woody detritus. 

I pour off the water one last time, still leaving enough that I'm not pouring out eggs, and swirl that and let it sit for a bit.  The swirling causes the eggs to collect in the center of the container where I siphon them up with a small syringe and transfer them to a shallow dish for drying.

I can repeat this process as many times as I want just by stirring up the gravel again.   Ordinary tap water can be added to the gravel if you run out of water in the tank.   In the space of a half hour or so I can easily collect several hundred to a few thousand eggs this way.

Once I'm satisfied with the haul I gently siphon off most of the water covering the isolated eggs and allow them to dry at room temperature for a few weeks before dislodging them from drying dish with a soft paint brush.  I then perform a rough and dirty count and mix them with my homemade detritus (next section) at a ratio of roughly 50 eggs per teaspoon of detritus.  I use one teaspoon of my egg mixture in approximately one liter of distilled water for hatching.

Note: First, once you've collected the eggs you can wash the gravel clean in a bucket like you would if you were setting up a brand new tank  This allows you to keep a cleaner triops tank while reusing gravel. 

Second, the transfer of the eggs to the relatively pure tapwater will trigger a small number of hatchings.  You can either let nature take its course or transfer the larval triops into another container of water treated with homemade detritus.

You can make your own detritus if the kit you bought doesn't provide one. But most of them do. it comes in a teabag or it is packed and mixed with the eggs

The problem of hatching triops is that the water must closely mimic conditions found in their natural habitat.  Without a suitable source of detritus or other water conditioning, larval triops usually die after a few days, often due to sharp increases in bacterial numbers or lack of food.  The most common way commercial kits provide a proper environment is through mixing the eggs with dried pond detritus which contains algal and protozoan cysts, provides disolved organic compounds in the water, etc.

Now, most of us don't have a pond in our backyard for harvesting natural detritus but we do have a pet store in town.  What I use for making homemade detritus is ground coconut shell (usually marketed as bark) which is sold in compressed bricks for reptile bedding.  This stuff is great, a single $6 brick will let you make enough detritus to hatch triops for the next decade (at least).  I mix the coconut bark with tank water from an established triops or other fish tank until it forms a thick mud.  I then dry it in a food dehydrator, but it could just as easily be spread thinly in a baking pan and allowed to thoroughly airdry.  That's it.

The tank water provides the protozoan and algal cysts.  The coco-bark leaches tannins into the water which both simulates the natural pool environment and inhibits bacterial growth.  It's also edible for triops which is a bonus.

Detritus Update:

Another method for creating detritus is to take some fallen dry leaves from a tree or strands of hay and soak them in old pond or rain water or existing aquarium tank water for an few hours and then dry them out again. This allows the infusoria which has grown onto your material to go into a 'hibernation' state ready to hatch out the next time you place the Detritus in water.

Break up the plant material into small pieces around 1cm in size as this makes it easy for the pieces to float to the sides of the tank and not take up too much surface area.

Tip: To make it easier to remove the detritus from your tank. Place it in a perforated bag. Herbal tea makers sell these types of bags or you could use muslin cloth or dish cloth stitched into a small bag.

I bought a pack of the cheapest dish cloths from a supermarket .

These are loose weave, double layered stitched cloth, so I simply cut one corner up 4cm x 15cm and stiched up one side leaving the top open.
I then added a piece of lead weight to help one end sink and after adding my leaves I closed one end by using one of those food bag closure clips you get from the supermarket. Very handy your local supermarket.

I then placed the bag into a old 750ml 'baby beetroot' jar, added some rain water, shook the jar, removed the lid and left the jar exposed to sunlight.

After an hour or so you'll see the water turn brown if you used old leaves, this is simply a non-toxic and natural Tannin plant compound leaking into the water.

After one or two days look closely at the jar and you should see tiny, tiny things moving about. Add this 'soup' (including the bag) to your hatching tank containing 0.5Litres of pure distilled/ionised water. Then add the eggs.

This way the newly hatched Triops will have plenty of creatures to feed upon.

Obviously the bag is no longer required after 3 or 4 days as the Triops have hatched out and have grown so big that the tiny Infusoria food source is no longer enough to feed them. But don't throw it out. Simply dry it off and you can use it again to hatch out your next batch!

What is Detritus for?

Detritus is the name given to organic waste material from decomposing parts of dead plants. This material when dried out after being soaked in pond or mature aquarium water will contain on it the eggs of tiny Protozoa creatures and algal cysts (tiny plant seeds) (these forms of life are collectively known as Infusoria).

These creatures hatch out once it becomes wet and along with particles of decaying plant material settle near the surface of the water.

Triops larvae being so small at birth, need an even smaller food source and move upwards towards a light source to find something to eat at the surface and in the wild they would easily find this natural source of food.
So in a bowl or tank environment we need to add our own.

If they don't find Infusoria or there isn't enough of it per squared centimeter of water then the Triops quickly die due to starvation.

Hopefully this section explains why you need to add Detritus to the water and why you don't use a large amount of water as otherwise you'd need to add a lot more Detritus to compensate.
A small teabag sized amount of Detritus, as supplied by Triops kits, is ample enough for 0.5 litres of water.