Why Mumbai Indians got that extra ball

The final score rather than the target score is taken into account for calculation of net run rate. Mumbai Indians needed to win in 87 deliveries, but that would have been the case had they scored exactly 190, the target score.
Mumbai Indians could have taken a few more balls to get their NRR to the required level and achieved a final score superior to the target score had they signed off with a boundary.
When they failed to scamper for two runs to get to 190 in 14.3 overs, and the scores were level, their net run rate of 0.078 was higher than Rajasthan Royals' NRR of 0.076.
Scoring just one run off the next ball would have seen their NRR slip below that of RR. Thus hitting a four or six for a win was the only option.
What is interesting though is that MI could have got through even by playing out a dot ball, but then would have had to hit a four or a six off 14.5 or 14.6 overs for their NRR to be superior to that of the Royals.
In fact, they could even have played three dot balls and could have gone through had they struck a six off the first ball of the 15th over. In that case, MI's run rate would have been 0.080519 against RR's 0.074163.