There is some controversy about this last part (often associated with differences over church governance).
However, there are two points that are clear and important.
One important point is the connection between the Apostles and the local church leaders.
Christianity is an incarnational faith: centered on the unique revelation and intervention of God in human history by our Lord and Savior Jesus Christ, the Eternal Word made flesh.
It is therefore quite fitting for those who are set apart for ministry to have a tangible, historical connection with the earthly ministry of Christ. (This aspect is often discussed in terms of "Apostolic Succession.")
The other important point (indeed, the most important point) about the ordination or appointment of these elders (presbyters) is that "they commended them to the Lord, on whom they believed."
On both points, the key thing to remember is that Christian ministry is to be grounded primarily not on the talents or ambitions of the individual ministers (even though talent and motivation may be desirable) nor on the politics of the people (even if some form of election may be involved).
Christian ministry is first and foremost founded on Christ and comes from Christ.
Christian ministry ultimately is not about him or her or you or me or them or even us: it is ultimately and thoroughly about our Lord and Savior Jesus Christ.