Why is the melting of ice a spontaneous reaction?
First of all, 


  what exactly is spontaneous reaction?


In very easy language, reaction that occurs in a given set of conditions without intervention is called spontaneous reaction. 



Now let us consider melting of ice example,

Take a ice on a plate and leave it for half an hour(conditions). After half an hour we will notice water instead of ice.

This process took place without any intervention from your side. Heat absorbed by the ice from surrounding is due to set of conditions not due to your intervention.

Thus we can consider melting of ice as a spontaneous process(or reaction).