entspringen
to spring
To emerge from the ground at a particular place as the source of a river or stream.
Der Rhein entspringt in den Schweizer Alpen.
The Rhine springs from the Swiss Alps.

to originate / to spring
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to spring
To emerge from the ground at a particular place as the source of a river or stream.
Der Rhein entspringt in den Schweizer Alpen.
The Rhine springs from the Swiss Alps.

'Entspringen' (to originate / to spring) is an inseparable verb. The prefix 'ent-' remains firmly attached to the stem 'springen' in all sentence forms.
Man weiß nicht genau, wo dieser Fluss entspringt.
People don't know exactly where this river springs from.
The perfect tense of 'entspringen' (to originate / to spring) is always formed with the auxiliary verb 'sein' (to be). The verb describes an origin or a movement out of something.
Die Donau ist im Schwarzwald entsprungen.
The Danube has sprung from the Black Forest.
In the most common meanings (source of a river, origin of an idea), the origin is indicated with the preposition 'aus' and the dative.
Der Bach entspringt aus einer kleinen Quelle im Wald.
The stream springs from a small source in the forest.
When 'entspringen' (to spring) refers to geographical places (mountains, springs), the literal meaning is intended. If it refers to abstract nouns like 'Idee' (idea), 'Fantasie' (imagination), 'Notwendigkeit' (necessity), or 'Gefühl' (feeling), the figurative meaning is intended.
Der Amazonas entspringt in den Anden.
The Amazon springs from the Andes.
The meaning 'fliehen' (to flee/escape) is outdated and is now used only in elevated or literary language. It is very rare in normal usage.
Der Ritter konnte dem Kerker unbemerkt entspringen.
The knight was able to escape the dungeon unnoticed.
In the meaning 'fliehen' (to flee/escape), the thing one is fleeing from is in the dative. One escapes 'der Gefahr' (the danger) or 'dem Gefängnis' (the prison).
Er war froh, dieser unangenehmen Situation entsprungen zu sein.
He was glad to have escaped this unpleasant situation.
to arise
'entstehen' and 'entspringen' can both express that something has its origin in something. 'entstehen' is neutral and very common in everyday language. It simply describes that something forms or newly comes into being as a result of something. In a figurative sense, 'entspringen' is more elevated and places stronger emphasis on the source or inner origin, for example an idea, a wish, or a fear. In normal conversations, 'entstehen' often sounds more natural. 'entspringen' is more suitable when you want to express the origin especially clearly or in a somewhat more refined way.

Sein Wunsch zu reisen entspringt einer großen Neugier auf andere Länder.
His wish to travel originates from a great curiosity about other countries.
Durch das Missverständnis entstand eine unangenehme Situation.
The misunderstanding created an unpleasant situation.