Beginner's guide to $A_{infty}$-algebras

I have some general questions about $A_{infty}$-algebras. Altough I
understand bare definition from nLab I have no association how to think
intuitively about them. Which picture one should
have in mind thinking of $A_{infty}$-algebras? What makes them interesting in higher algebra & theory of spectra? Which questions from these areas they primary help to attack? Hope, it’s not to broad formulated.

How they are concretly related to a infinity loop spaces? Can they be thought as a natural homotopy theoretical pendant of infinity loop spaces? (i.e. that $A_{infty}$-algebras are related to infinity loop spaces like homotopical colimits to usual limits? Or is this a bad intuition?)

In this MO question Naruki Masuda wrote an interesting comment I would like
to understand:

By considering loops parametrized by [0, t] for all tgeq 0, you
see that a loop space is a deformation retract of a
(strictly unital and associative) topological monoid.
But a strict algebra structure is not preserved by homotopy
equivalence, so topological monoids are not a correct notion
of ‘homotopical associative monoid.’
The definition of A-infinity algebra, which is the homotopically
correct intrinsic notion of associative monoids, takes a cue
from the recognition of algebraic structures present in loop spaces.

I would like to elaborate precisely how the
definition of A-infinity algebra arise precisely from
recognition of algebraic structures present in loop spaces.
Can this a bit "hand weavy observation" be made more precise?

Additionally, I heared that $A_{infty}$-algebras become quite
important in physics (TQFT). I’m
looking for a sketchy
motivation how these structures in theoretical physics naturally occure.