Other than the constitutional role of presiding over the Senate, the primary role of the Vice President appears to be the promotion of the President's agenda. The Vice President sets no policy, cannot veto any law and, in the Senate, can only vote to break a tie. Those who accept to be Vice President may do so to improve their own chance of a future run for the Presidency, but they understand that they will not be advancing their own agenda except where it aligns with the President or to whatever limited extent they can in their role in the Senate.
The choice of running mate is, in my opinion, made for two reasons. One is agreement on political agenda. Since the Vice President will spend a significant time promoting and supporting the agenda of the President, there must be at least significant agreement between the two in regard to agenda and policy. The other reason is the need to win the election in order to even be able to move forward with that agenda. In other words, one of the reasons behind the choice is the hope that the running mate can make up for weaknesses in the electability of the candidate for President. In order to win an election, the candidate for President must be seen to support the major voting blocs within their party's supporters - those members of their party's base without whom they essentially stand no chance of winning.
Two of the largest voting blocs on whom the Republican Party's candidate depends are Evangelical Christians and the pro-life movement. These two blocs are mostly, but not completely, the same within the party. In recent years, with growing public dissatisfaction with government, candidates must also be able to portray themselves as being against the status quo or as a political "outsider" and be an advocate for "smaller" government. Donald Trump has received criticism by conservatives regarding his position on abortion, and even some on his understanding of Christianity. His choice of Mike Pence has been viewed as a confirmation that he is aligned with the Evangelical Christian and pro-life blocs of the party.
The Democrat Party has several large voting blocs which, like those in the Republican Party, overlap but are not the same. Two of these groups are the social justice movement, which focuses much of its criticism on big business and war, and the pro-abortion movement. A very large portion of the social justice movement within the party is the Catholic vote, which could be considered its own voting bloc in the Democrat Party. Despite the party's unwavering support of abortion, its perceived support in other areas of social justice appeals to many Catholics and several Democrat victories would not have occurred if they hadn't received a significant number of votes from Catholics. Recently, Mrs. Clinton has not only been increasingly viewed as a "war hawk," but also as being in league with big business interests opposed to the very things the social justice movement supports. It should also not be assumed that, just because the Catholics in the party are willing to side-step the issue of abortion for other social justice issues, Mrs. Clinton's views on religion won't hurt her with that bloc. Beyond the fact that she asked who painted the tilma of Our Lady of Guadalupe (an incident that can be blamed more on her staff than on her), Hillary has made generally anti-religious statements including saying that deep-seated religious beliefs must be changed. These statements won't sit well with many Catholic Democrats. If she hopes to win, however, anything she does to gain the support of Catholics in the party can't upset the pro-abortion bloc. Her choice of Tim Kaine walks the thin line between these two blocs.
The choice by the candidates for President of who will be Vice President is frequently viewed as a "sign" among the party supporters revealing more about the "real" positions of the presidential candidate. Voters should examine the significance of these choices. Just how much does the choice represent the presidential candidate's own views, and how much is it an attempt to appease those portions of the party who wouldn't otherwise want to support the candidate? Does the candidate for Vice President represent the presidential candidate's own position, or is it an attempt to get the voters to overlook where the candidate's views don't really align with their own?