More than you might think - but not as much as some people claim
This is an excellent example of the worst kind of bad science: it jumbles up truth with untruth, and you end up throwing out the baby with the bathwater. The truth lies somewhere in the middle ground, but mixing it up with bunk only discredits everything.
It turns out that plants do sense and feel and react to their environment in surprisingly animal-like ways. They do react with faint electrical impulses, chemical pathways, which can be invoked if you listen closely enough. An artist named Miya Masaoka created an art installation where she hooked up a bunch of plants to electrodes, and output the sound through a speaker. When you touch or brush the plant, it creates an electrical reaction, which allows you to play the plant like a musical instrument of sorts.
But plants can't psychically empathize with a pot of dying shrimp in the kitchen.
Part of the problem with relating to plants is that they don't live at the same speed that we do. (A very fine Star Trek Next Generation episode was built around this very idea.) Plants actually react to their environment more than some things we call "animals" do. Jellyfish, for example, are considered animals, but they seem to exert less volition than a group of trees.
When threatened, a tree will push out a chemical warning to the other trees in its stand, warning them of the danger. This gives the other trees a chance to mount a chemical defense against the invader, and to tailor it to the threat. A tree will release certain chemicals in response to a fungal threat, and other chemicals in response to an insect invasion. And it will do so based on the word of a distant tree which is serving as the sentinel.
The Jains famously restrict their diet only to vegetable products that won't kill the mother plant. Onions and potatoes are out, because it kills the plant to harvest their roots. Apples and oranges are in, because the trees drop these without consequence. Maybe they are on to something!