Ok first of all, go to the part where Neo is in the car with Trinity, Apac, and Switch going to meet Morpheus.
When Trinity takes the bug out of Neo's naval with the machine, you can see that the bug has a string tied to it! What a pathetic slip-up, the movie makers could have done better than that...
It's just a stream of the bug's entrails sucked out by the vacuum created by the machine. Now don't start arguing about how the bug isn't an actual insect and actually a little mechanical device, you all saw the device obtain flesh.