Why does this script return an error ? I'm trying to understand how Script validates transactions. 

I have taken this transaction as an example 55e5bbdba33050bbe6584e896b4a20575cf3fe3f29114467f17f67f6b172eb73

When showing the scripts, we can see as input

ScriptSig: PUSHDATA(72)[3045022100c70904b2925a6822fd3dc2d236bcba1fa1518fdfdadbda8f25a89a339879c82a02200bd7ca7d502d46ebc6fbc97f51af3d0ab4f290d73931ada9abb76fb1b6d3b55b01] 
PUSHDATA(65)[043fd05d2490b2b8496da3b8498f1aee08634c43eef40aa58b66e36271632571d6dbda0ec7fb6fcec09409597e38e3b968648ed10ad6402647bb6143f0f9f09b3f]


And output : 

DUP HASH160 PUSHDATA(20)[57d713220076f8298b54e9eea84d2e1dde62e4a1] EQUALVERIFY CHECKSIG


So, from what I understood, if these script bits are run one after another, the result should be true (please correct me if I'm already mistaken at that point).

I have used an online Script debugger but these bits did not return a success. 

Has anyone an explanation on why I couldn't get this right ?
You are not supposed to be using the output scripts of the outputs created in the transaction but rather the output script of the output that is being spent. What you are using is the output script of the output in that transaction.

Rather you should click the output link next to the input and get the output script from the referenced output that the link takes you to.