Summary: Examining the responsibility to seek out and prepare faithful men who will advance the Faith in days to come.
“You then, my child, be strengthened by the grace that is in Christ Jesus, and what you have heard from me in the presence of many witnesses entrust to faithful men who will be able to teach others also. Share in suffering as a good soldier of Christ Jesus.” 
Apostolic succession is essential to a healthy church. However, whenever we speak of “apostolic succession,” it is important to define what is meant. When I speak of apostolic succession, I imply adherence to the model provided in Scripture. I do not refer to the aberrations, however popular they may be, that occur in contemporary church life. By “apostolic succession,” I mean that there is a continuation of the Apostles’ doctrine.
Our Catholic friends, Orthodox Christians and numerous Protestant communions as well, hold to what they call “apostolic succession.” By this term these religions infer that they depend upon being able to trace the ordination of priests back to Peter. Tragically, we are aware of the scandals that have marked many of the churches holding to this particular aberration. Those who perpetuated and attempted to cover over crimes which were committed by priests were not at all living as Peter would have lived. It is obvious that God could not approve of their actions. They would argue, however, that because of a succession of ordination back to Peter, then doctrine and full fellowship are ensured. 
Among some evangelical churches, and especially touted by one particular Baptist group, is held a form of apostolic succession that has caused great harm to the cause of Christ. These churches do not argue that they have a succession of ordination; rather, they hold to “baptismal succession.” Their argument is that your baptism is invalid if you cannot trace your baptism back to John the Baptist. Their de facto argument is that only baptism administered by a church holding the name “Baptist” is valid; and all other baptisms are, by definition, defective.  My wife is an excellent genealogist; however, not even my wife is able to trace such a lineage.
Apostolic succession, however, is readily witnessed by appeal to the Word of God as we hold churches to the mirror of the Word. You will note a statement concerning the practise of the first congregation in Jerusalem. “[Those baptised] devoted themselves to the apostle’s teaching” [ACTS 2:42a]. These earliest Christians were committed to knowing and obeying what was taught by the Apostles. That mark is witnessed to this day among churches that practise, whether they use the term or not, apostolic succession. Apostolic succession is the succession of teaching that defined the first congregations and continues to mark those faithful congregations to this day.
In our text, Paul clearly tells Timothy that he is responsible to hold to what he has been taught. Then, he is responsible to ensure that this truth is passed to others who themselves will faithfully hold these same teachings. In this way, continuation of the apostles’ teaching is assured; this ongoing communication will transcend time, culture and language. If we question whether the things taught from the pulpit align with what the apostles taught, we need but review what is recorded in the Word of God to give ourselves assurance. Whilst the continuing communication depends upon those proclaiming the truths, those listening bear responsibility to know the Word and to hold what is taught to the standard of the Word.