In Jain practice, the sound or word you hear as you leave isn’t “asihi” exactly; many say “Nissihi” (Prakrit) or “Ävissahi” (Sanskritized form). Both mean renunciation and are used to turn the mind away from worldly thoughts as you depart the temple.

• Why we say it: It is a moment to renounce attachments and keep the darshan (worship) in the heart. Saying Nissihi helps fix the intention on the Jina and sacredness of the place, even as you step back into daily life.
• When it’s done: In the Shwetambar tradition, Nissihi is recited as part of the temple protocols, typically at the entrance, near the sanctum, and before starting the puja, i.e., a form of renunciation in three steps. The key idea is to leave worldly thoughts behind and remain focused on the Jina. ( jainknowledge.com)
• Different names/renderings: You may also hear it called Ävissahi or similar forms; all point to renunciation and turning the mind toward darshan. Both Digambar and Shwetambar traditions emphasize renunciation, though the exact phrasing and placement can differ.

If you’d like, I can point to a detailed explanation from Jain Knowledge that explains the three Nissihi steps in the Shwetambar practice.