Do I need mana to play an instant that lets me block my opponents attack?
Yes, you need mana in order to cast spells that require mana. You can't just cast Spidery Grasp for nothing; it costs {2}{G}. If you have no mana available, and all your mana sources aren't available to be used (e.g. all your lands are tapped out already), you will have no mana to cast it with.

Lands and cards like Llanowar Elves can be tapped to give you mana, but won't give you mana merely by existing.

I suggest you give the game's Quick Start Booklet (PDF link) a read. It might clarify things for you.