Why triplet excited state is found to be more stable than singlet excited state?

This electron is in an excited unstable state.

Yes, because an atom in an excited state will normally give off energy and go to a less-excited state or to its ground state. Some atoms have long-lived excited states and are called "metastable".

state in which electrons have absorbed energy and "jumped" to a higher energy level

When the electrons are at higher energy level,they are said to be excited state.