In the calculation of net run rate (NRR), the final score, and not the target, is the relevant number. For Mumbai Indians, the requirement to finish the chase in 87 deliveries was only subject to their final score being 190. The chasing team, if they take a few extra deliveries to get home, can still push their NRR up to the required fraction if they manage to achieve a final score that is sufficiently higher - by finishing things off with a boundary.
Mumbai Indians, despite failing to score that all-important extra run off 14.3, had already inched ahead of Rajasthan Royals' NRR when they had drawn level on 189. At that stage, Mumbai Indians' NRR read 0.078099, while Royals' was 0.076821. Had Mumbai Indians just run the single they needed for victory off the fourth ball, though, their NRR would have gone below that of Royals'.
Since they were using the extra ball, they would have needed to get their score up to at least 191 off that delivery. Running two was not an option, as they needed just the one run to win. So their only option was to hit a boundary.
However, had they played out a dot ball, they still would have not been out of it. They could have hit a four off 14.5 or 14.6 and still finished with a NRR better than that of Royals. If they played out three dots, they would have needed to hit a six off 15.1 to bump their NRR up above Royals'. If this had happened, Mumbai would have ended with a NRR of 0.080519 against Royals' 0.074163.