Historically this selection has had only a small effect on the race, but by choosing a vice president, the candidate is making clear who they think should be a key advisor and worthy successor. It may not have been the goal, but modern-day vice presidents have become a crucial member of the president's policy team - the only one he cannot actually get rid of - and by far the most likely potential successor for the party's nomination.
While vice presidents were historically laughing stocks, over the last half century the position has changed. For 124 years, from 1836 until 1960, only seven vice presidents ascended to the presidency and they all did so based on the death of a president – and four of those seven could not even get their party's nomination for the presidency after they moved up. Not one vice president received their party's nomination for the presidency at the end of their term.
However, starting with Richard Nixon, seven of the last 12 vice presidents went on to win their party's nomination for the presidency in the next election. The five who didn't get the nomination all have their own reasons - Spiro Agnew, who resigned in disgraced; the unelected Nelson Rockefeller, who died before the next election; Dan Quayle, who never escaped ridicule; Dick Cheney, who had severe health problems; and Joe Biden, who didn't run this time due to Hillary Clinton's ability to cloak herself in the mantle of both of the last two Democratic presidents.
Biden's difficulty in mounting a primary race against Clinton points out why the VP is so important. Biden could have run, but he clearly thought that Clinton had established her credentials as Obama's and her husband's preferred successor. The reality is that in modern times the running mate is seen as the heir apparent. Any president leaving office still possesses a large degree of popularity at least with his or her own party's voters, enough that the candidate seen as close to the party leader has the best chance of grabbing that nomination. That person is generally the vice president. And presidents want their vice presidents to move up. This succession is a ratification of the president's own term.
There are some real reasons for the recent change in VPs succession ability. During the political convention era, presidential candidates were selected by state leaders and machine bosses. The vice president was chosen at the last moment, and it was used a bargaining chip to get the presidency and a way to provide a consolation prize to the losers. The result was that many of the men who were taken as VPs were neither prominent nor powerful.