How Can I fix a broken hose?
There are garden hose splice devices that you can purchase at home centers and big box stores that can be used to fix these. You may lose a short bit of total length as you cut off the damaged part to accomodate the splicer.



If you need more added length you could use two of these splicers to stitch in a new length of hose. 

They also sell devices that will attach new hose ends as well. 

In your situation the hose ends that are broken may be broken from having laid out in the sun for years at a time. The sun treatment can make the hose hard and  brittle. On the other hand the part of the hose in the soil may still be in decent shape. You could cut off the old hose all the way back to where it is in the soil and splice on a piece cut from a new hose that already has the faucet tap fitting on the end. 

Be aware that the direct attachment of your lawn sprinklers to the the potable water supply is not the recommended guideline. Sprinklers are generally connected through anti siphon valves (whether those be manually operated or solenoid controlled). You may want to look into replacing the temporary hose connect hack with a more permanent arrangement with some anti siphon valves. Note that the anti siphon valve should be installed at an elevation that is 8 to 12 inches above the highest sprinkler outlet on that water circuit.