What is the purpose of making a pilgrimage to Mecca?

to study the Five Pillars of Islam

to pray at the site of the Ka'bah

to tour the birthplace of Muhammad

to visit the shops of wealthy merchants

The purpose of making a pilgrimage to Mecca is to make Hajj, which is one of the Five Pillars of Islam. You pray at the site of the Ka'bah and Mecca is also the birthplace and deathplace of Muhammad. So the second option :)