So then, how does this concept of the tituli relate to the cardinals? The concept of a "cardinal" (a word that has its origins from the Latin word for "hinge" referring to something or something of importance) comes in relation to those who made up the chief administrative body of the church within Rome; counsellors and assistants to and under the authority of the bishop of Rome -- that is to say the Pope. Cardinal-deacons were given the responsibility of the "diaconiae" (which were the charitable outreaches within the city) while the Cardinal-priests were given responsibility over the tituli -- what today we would think of as parish churches. In a nutshell then, the tituli represented the various centres of worship within the city of Rome and each were assigned their respective "cardinales" -- in modern terms, effectively the parish priest or archpriest. When a cardinal is assigned and takes possession of his titular church in Rome what he is therefore doing is symbolically being made the titular (i.e. primary) priest of that church within Rome.
As the Church grew, and so with it the role and function of the cardinals in the administration of the Church in communion with the pope, this act has become more symbolic today than it is actual, however when one understands the historical background of how it relates to the primitive organization of the church in Rome, one can better understand the meaning and symbolism of it.
It is worth noting that not every church in Rome is a titular church. Churches are designated as a "titulus ecclesiae" by authority of the pope who can (and who historically have) also revoked such a status.
So then, in summary, just as we today are familiar with the idea of a "cathedral" as a designation that denotes the location of the seat (i.e. cathedra) of the local bishop, or that a "basilica" as an honorific title bestowed on particularly beautiful and important churches (frequently but not necessarily cathedrals), so too is a "titulus" a title that comes with specific reference to the most important churches within the city of Rome, to which each is assigned a cardinal.
The ceremonial act of a cardinal taking possession of his titular church can be seen below in these photos from 2011 when Cardinal Raymond Burke took possession of his own titular church, Sant'Agata dei Goti.