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I can kind of understand how Counselor Troi could sense the emotions of people in the room with her, that is at close distance.
But in communications with other ships or planets, she was able to sense others' emotional states. (She could even sense the general state of the crew, but that's probably a separate question.)
Was there any range limit? Is her super-power dependent on something physical, which would imply a range limit?
Telepathic/empathic energy is an actual force in Star Trek - it has ranges, strengths, and is affected by various phenomena.
One example of extremely strong telepathic energy - so strong it prevented the entire crew from entering REM sleep - came in TNG 4x17, Night Terrors.
Related is Psionic energy, which if I remember correctly, Deep Space Nine's sensors were able to detect after calibration.
Just like a Faraday cage will block an electric field, and other substances provide resistance, it seems easily plausible that psionic energy and telepathic/empathic energy can be affected by the physical universe similarly.
And yes, as mentioned in @user13095's answer, we know for certain there is a general range limitation to Betazoid empathic powers, based primarily on each person's own skill: in TNG 3x20, Tin Man, Troi is shocked to learn how far away Tam Elbrun was in contact with "Tin Man".
I watched the entire series not too long ago, I do not recall any sort of limit ever being stated. In terms of physical distance, it seemed to be whatever was required by the plot. Also, she was only half Betazoid, her mother had significantly stronger powers.
However...I do remember an episode called, Tin Man? I might be wrong on the title. At any rate, it involved a sort of prodigy Betazoid who was in touch with another being over light-years.
Upon entering the Rana system in "The Survivors", close enough to Rana IV for a visual display of the planet's surface, Troi says, "There are 11,000 inhabitants of this colony. At this range I should be sensing something; I'm not." As the Enterprise approaches the planet, Troi is able to sense the two "humans" on the planet. Later, when the ship is in orbit and without going to the planet, she knows that there is something "different" about the two beings on the planet. Clearly, Troi's empathic abilities are affected by both distance and the number of people she is sensing.
