How do mountain springs get their water?
Ultimately, it comes from precipitation.  Ordinarily we think of rain as coming from low-level clouds, but Putkonen[1] has compiled rainfall data in the Himalayas showing significant rains up to several thousand meters altitude, covering the range where practically everyone lives.  It is this precipitation that fills the underground tables mentioned by Jean-Marie Prival in a comment to the question.

Such a source is subject to the effects of climate change, which accordingly has led to significant environmental issues.  See Ref [2].

References:

1.
Jaakko K. Putkonen, "Continuous Snow and Rain Data at 500 to 4400 m Altitude near Annapurna, Nepal, 1999–2001", Arctic, Antarctic, and Alpine Research, 36:2, 244-248 (2004)

2.
Sandeep Tambe, Ghanashyam Kharel, ML Arrawatia, Himanshu Kulkarni, Kaustubh Mahamuni, Anil K Ganeriwala,
"Reviving dying springs: climate change adaptation experiments from the Sikkim Himalaya",
Mountain Research and Development 32 (1), 62-72 (2012)