What are effects of attaching more antennas to Wi-Fi router via RP-SMA splitter

At best it probably won't help and at worst it will degrade your signal if they are setup like in the picture.

Imagine an incoming transmission coming in parallel to the line the two antennas create. The signal is picked up by the first antenna then travels some short distance where it is picked up by the second antenna. That signal will now combine at the splitter out of phase and likely cause destructive interference.

I cant say for certain but I'm assuming the two or three typically attached to these routers are designed at the system level not to interfere with each other.

Some systems I have seen use a coupler with antennas pointed in different directions but share a receiver. With this approach you'd need to find the radiating pattern of your antennas and used short coax to arrange them so only one picks up a signal for a given transmitter/receiver geometry. As Evan pointed out a hybrid coupler would be used. The application in the link is for sharing one antenna but covers the basics

Two parallel antennas means impedance mismatch and lower power to both antennas in the very best scenario. Closely positioning the antennas can mean a radical reshaping of the emission and reception of radio waves. In the worst case scenario you might damage your radio device due to power reflection because of the impedance mismatch.