I see the address 1Ahw6xqEmarByEaCeVKitLQfprHhEqRjQW1Asw9BGefMxkG4TTUhs1fsiKuyCQRmeccy in my Electron cash wallet now.
I see the address in the Addresses tab but what do I do with it now? I do not see a monetary value associated with the address. How do I know how much value is actually on this address?
There is nothing in my Coins tab.....
Then your client probably isn't properly snyced (check the circle at the bottom right, it should be green - if it is red, you might connect to another server to retrieve the information).
Your address
1Asw9BGefMxkG4TTUhs1fsiKuyCQRmeccy does hold a balance of 0.271.. BCH.
Your address
1Ahw6xqEmarByEaCeVKitLQfprHhEqRjQW contained 0.271 BCH until you sent them to your other address (1Asw9BGefMxkG4TTUhs1fsiKuyCQRmeccy) on 2018-09-24 at 20:30.
You can always check your balance on a block explorer.
https://blockchair.com/ for example does support BTC and BCH.