To substantiate is to give support to a claim. We’d really like to believe in the Tooth Fairy; however, more evidence is needed to substantiate her existence (besides that quarter in your pocket).
Substantiate is related to the word substantial, which means "solid." So, to substantiate a claim is to make it solid or believable. If the evidence given in support of an argument is weak and unconvincing, that evidence can be described as insubstantial. Of course, in special cases like the Tooth Fairy, having substantial evidence doesn’t seem to matter; fans just keep on believing.