How was Harry able to find out Snape was on the good side

Towards the end of the book 'Harry Potter and the Deathly Hallows', Voldemort orders his snake Nagini to kill Snape. As Snape is lying on the floor taking his last breaths, he gives Harry memories that relate to his protection, conversations that Snape had with Dumbledore and memories showing how Snape fell in love with Harry's mother. Harry then viewed these memories in the pensieve and was forced to realise that Snape had been attempting to protect Harry and was acting as a spy for Dumbledore all along.