Craftsman Eager1 6.75hp mower

How long does it run? If only a few seconds, then it is running on the fuel from priming, and the carb is not supplying fuel to keep it running. If it runs for a minute or longer then stalls, then you have a restriction or blockage in the fuel lines, the tank, or the area where the fuel line connects to the carburetor.

You didn't mention which engine it had. Most carbs will have a round bowl on the bottom, held in place by a brass nut (often 9/16" wrench size)
Remove the bottom nut on the carb bowl. Depending on the carb model, the nut may have some holes on the inside--these are part of the metering system, so don't damage the holes or scrape brass away as you clean out any scale, corrosion, etc.

I find that a piece of 14 gage copper automotive wire works well. Remove the insulation about 3/8" and use the wire strands like a brush. Also run a strand through the holes as well as down the center of the nut.
Spray everything off with carb cleaner. The WalMart Super Tech brand is cheapest and works fine. Brake cleaner will work, too, but it doesn't have the lubricants that carb cleaner has.

Make sure you spray up through the bottom where the carb bowl bolts to the carburetor.

If you have knurled screws on the side or top of the carb, these are the needle valves. BEFORE YOU REMOVE THEM, carefully screw them in, counting the number of turns. Don't screw too tightly, just until slight resistance is felt. Then, screw them out. Spray them with cleaner and wipe down. Also spray the holes they came out of. (Make sure you don't lose the small springs under the screw head--they keep the screws from backing out when the engine is running).

Assemble everything again. Screw the needle valves all the way in, then back out the number of turns you counted when first removing them.

Make sure your fuel tank is clean. Moisture from condensation can collect--clean it all out. Blow out the fuel line with compressed air--even an inflating needle on a tire pump will work if you seal the end of the hose with a rag after inserting the valve. Give the pump a couple strokes and you're done.

Now start the engine. It should run fine. If it doesn't, take it to a small engine mechanic.

It may have a crack in the fuel line from tank to carb or the plunger may have a hole in it it all works on gravety feed & no air leaks from gaskets & fuel lines hope i helped if not e-mail me & let me know what other problems it has thanks wade

Most of the starts but dies problems are caused by dirt/water/debris in the carburetor float bowl.

You can try draining the gas out of the carburetor float bowl by removing the float bowl nut located on the bottom of the carburetor.

Make sure the hole in the float bowl nut are clean. If the problem persists after draining the float bowl nut you will have to remove, dis-assemble and clean the carb with compressed air and carb cleaner.

Sounds alot like my mower. Same model and problem has been - runs rough, when starting from cold, it quits after a few seconds. Took mine to a great mechanic. He found the compression cycle was offset due to poor spacing in the intake valve. Ground a little metal off the end of the valve stem and now it runs most excellent. May not be the same solution for your problem, but it worked for mine.