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I can't judge the reliability of <http://www.kanjinetworks.com> (and it is
probably controversial), but there the speed, movement meaning of 逸 is linked
to its claimed origins in 兎{うさぎ｝. Hence 逸する, 逸らす and such that involve some
form of movement.

However, there are also words like 逸楽, 安逸 that are linked to pleasure,
idleness etc. Where does this meaning come from?
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etymology
How did 逸 come to feature in pleasure/idleness words?
逸 itself means "escape" or "break loose". One one hand, this can be used for
verbs such as 逸する, and on the other hand this "escape" has been taken in a
more literary sense of "breaking loose", which it's not hard to see how it
becomes "relax" or "leisure". This same meaning is present in Chinese.

**Breaking up the character**

As for the origins, 兔 is the original kanji for ウサギ, with 兎 as a variant.
Standardization in Japan for some reason picked 兎 for ウサギ rather than the
first one, while leaving 兔 present in 逸. The character is made up of parts for
"movement" and "rabbit / hare", both acting to show the meaning of fleeing.

**Kunyomi and finding source of this meaning**

A number of the kunyomi reflect the meaning when it was loaned to Japanese.
The kunyomi are: はしる, うしなう, それる

Granted, I've never seen はしる written with this kanji, but it was loaned as a
reading (showing similar semantic use in Japan when inherited from China).
Hashiru obviously means "to run" (written with 走 now), Ushinau means "to part
with", and soreru means "to go astray".

**So where did it come from?**

The meaning of "relax" is used in Chinese compounds, and in none of the
kunyomi Japan gave the character initially, which suggests that at the time
Japan started using this kanji it didn't have that meaning, but came to mean
"relax" in China, and so the additional meaning of "relax" came along in
compounds such as 安逸 and 逸楽 in Japan.

tl;dr came from a development in the Chinese language sometime after kanji
were first introduced to Japan.
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Kanji dictionaries (漢和字典) are what to look up when confronting questions like
this. From what I've got from my dictionary:

  * My dictionary states 逸 actually is linked to 兎 in origin: "rabbits run away" -> to run, to astray.
  * For 安逸 and others, it states that they are used (in modern usage) in substitution for "佚", whose phonetic value is also "いつ". So 安逸 was originally 安佚．"佚" has no kunyomi, and its meanings include "たのしむ (to enjoy/have fun)", "なまける (to be idle)". 佚 comes from 人+失, and according to the dictionary, the origin is "まともな生活からそれたひと", so perhaps it is a good substitute even when considering the meaning.
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