And seriously, I am puzzled. Does no one sign transactions offline anymore?
I do that (using my own tools) - so as you asked I'll try to explain the usage of http://ciyam.org/rawtx_helper.html for you (which might help with your usage of other such tools as well).
Firstly you need to understand that the key thing about being sent bitcoins is that "you are not being sent coins" but instead what are called UTXOs (Unspent TX Outputs).
You could perhaps consider these as being more like "cheques" than coins (in that the value of them can be any value but not in the sense of having to worry that they might "bounce").
To give you some illustration of what I mean please look at the following: https://blockchain.info/address/1ciyam3htJit1feGa26p2wQ4aw6KFTejU (these are the txs from the 1ciyam address in my sig).
So if you had been sent "coins" to an address then you could type that address into a website like blockchain.info to come up with a similar looking set of txs.
The next thing is to look at a particular tx such as this one: https://blockchain.info/tx/135e55010a9caccd412d2c793a18236e9eebbb38412e97dca2b09aa8c354e1c0 and to see the full details there is a "Show scripts & coinbase" link you can click on (on the RHS - last line of the Inputs and Outputs table).