The logic behind having an emergency fund is this: if you lose your job, or for whatever reason are unable to work all of a sudden, or suddenly have a medical emergency on your hands (your own or your parents', spouse or child's) that may or may not be covered by insurance, your life shouldn't come to a grinding halt on the financial side.
You should have enough money in the bank (for immediate access) and in liquid funds to meet the emergency related expenses and also to help your life continue on its normal course until you can get back to work.
As you get older, the likelihood of this kind of medical emergency happening goes up.
You don't want to find yourself in a situation where you suddenly need Rs. 10 lakhs for an unexpected surgery, need to take 2 months off work for recovery, and are strapped for cash.
The last thing you want to worry about at that time is 'My SIPs are all going to stop due to insufficient funds', or 'The cheque is going to bounce because my salary is going to be affected'.
This is the kind of thing that happens to people who don't plan ahead, and you don't want to be in that situation.
You need to have a contingency fund in place that is the equivalent of at least 6 months of regular cash outflows.