Newly hatched BBS will be too small for adult bettas to eat; they are roughly smaller than a pin head - they are used for newborn fry. You will have to grow the BBS up to adults - and make sure not to feed the egg/shell casings.
You can use a brine shrimp net or a turkey baster to collect the brine shrimp, then under gentle water just rinse them under the sink faucet (while the shrimp are in the brine shrimp net) and then put them into the tank to feed. Make sure not to get too many at once though..
They can survive for about 4-5 hours in fresh water.
Use a BBS hatchery to hatch the eggs.. To grow your brine shrimp onto adulthood, you must feed them. For the first 24 hours of life your brine shrimp will not require feeding as they are still relying on their egg sac, and have not fully developed their digestive system. Brine shrimp are non-selective filter feeders (meaning they're not fussy) and can be fed a wide range of foods, including but not limited to Liquidfry brand or similar fry food, yeast, whey, wheat flour, soybean powder, fish meal and egg yolk. The criteria for suitable foods is small, easily digestible particles which do not dissolve in water (for example, powdered milk won't work). Most importantly, do not over-feed them.
It will be necessary to change the water in the hatchery regularly. An easy way to do this without losing any of your tiny brine shrimp is to siphon the water out through an airstone. Replace the siphoned water with adequately salty water (just make a guess based on how much salt you originally put in, in comparison to how much water you took out - remember too much salt is better than too little.)
Depending on water conditions, foods and the strain of brine shrimp you are raising, it could take anywhere from a week or so to a month to reach adult size. How long, or more accurately, to what size you raise your brine shrimp to is up to you.
Brine shrimp need a clean environment and using the bottle they were hatched in won't be clean enough. You need to remove all the dead eggs, empty shells and any dead brine shrimpettes. You also would need to cull the hatch to give each remaining nauplii enough room to grow and have enough food to live on. They will need to be kept warm and water will need to be continually circulated. They also need a dark period (like day and night). It also takes most 6+ weeks to grow to adult size and usually most will die before then.
People tend to use larger tanks/buckets for growing these things.. and even with the right set up they are fragile and not many survive to adulthood.
Use a air stone for aeration cause the baby brine shrimp will stick to the tiny bubbles and die.
Brine shrimp are attracted to light. Using a light to attract them into one area can greatly aid in catching them. Without a strong light-source and in the absence of water agitation, brine shrimp naturally sink, while their hatched eggshells float.
This site may help you some