The book, although it was written in 1934, continues to be influential especially in art education circles. The Stanford Encyclopaedia of philosophy provides useful background and summary of key ideas.
Dewey deals with art in general using examples from painting, sculpture, and poetry to illustrate his ideas.
For Dewey, art is not about the object (painting, song) or about the artist who created the object. It is about our experience of engaging with art.
These contrast with two prevailing modes of analysing or discussing art: (1) exclusive focus on the art object (whether a painting or a song) and (2) exclusive focus on art as emotional communication (Dewey refers to it as emotional discharge from the artist)
The experience in the title relates to everyday life - we engage aesthetically with everyday objects and actions. But Dewey highlight that art brings different qualities to these “...art has a unique quality, but that it is that of clarifying and concentrating meaning contained in scattered and weakened ways in the material of other forms.”
He also thinks there is a unity in an artistic experience which is not present in everyday experience. This unity is both about a sense of closure or fulfilling internal trajectory and about the relation between parts and whole. The parts are working together within the whole but without being subsumed into it and, therefore, loosing their characteristics.
An experience is not passive but active. Dewey describes that as a doing and undergoing in engaging with art. To him this applies to both the artist creating the work and the observer engaging with the art. (And we can see echoes of this in work such as Noë's Action in Perception). This also means that all art (including paintings) involves time since it is a process of engaging.
An important aspect is the relation between the art (painting, music) and the medium (colours/shapes, sounds). While all art shares aspects (stemming from 'an experience') each has an individual language that depends on the particular medium.
Psychological mechanisms - from perception to emotion and thinking - are integral to art. From detailed observations when absorbing art to the emotional and intellectual engagement and the awareness (and self-awareness) that art engenders.
Expression, for Dewey, is also a process. He distinguishes between an momentary impulse (e.g. venting anger) and a more deliberate process of expressing oneself in art. Emotional states can be the starting point but self reflection and shaping of material (colour, shape, sound, gesture, words) can result in expression.
Just as Dewey is shifting his focus away from the art object to the process of engaging with it, he tries to describe the process through which artists shape the object.
Thus there is a symmetry between the creator of the artistic object and the observer: an extended engagement that involves emotional, intellectual, and embodied engagement.
In discussing the creative process, Dewey doesn't subscribe to the notion of revelation (that the artist is assembling materials to match an art object they have in their mind). The process is more one of discovery and gradual shaping of material.
By no means a real analysis but an illustration of how influential Meyer's book is:
Also compare Meyer's account of emotion and music with this lecture from Steven Brown: