Why it is important to find largest prime numbers?

Reason number 1: scientific curiosity.

Reason number 2 (the economic reason): The RSA encryption algorithm requires large prime numbers to make secure data transmissions. The larger the primes, the more secure the transmission.

Just to add to the previous answers: Usually, part of the discovery of these mathematical curiosities is not the result itself, but the new or improved method for finding these new results. It's not just about finding the number which is meaningless in itself, but it's about showing that mathematical techniques have advanced so much that we can even show that these incredibly huge numbers are prime.

Similarly, who cares that we planted a flag on the moon, or that we sent people there at all? It's all about showing that we can do this.

Because they're there.

Euclid called our attention to the fact after "because" in the third century BC, and Hillary called our attention to the appropriateness of "because" in the 20th century AD.

Also, large primes play a role in cryptography, which protects your bank P.I.N. number from thieves.