I guess the key to understanding this is to understand the word "elevated." In this case, it means "superior" or "better than the normal run of things."
So the quote you cite is, presumably, from someone who thinks quite highly of poetry. The author of the quote is saying that poets have thoughts superior to those of other people. The author then goes on to say that poets express those superior thoughts through the use of superior language.
So basically your quote is saying that poets are better than everyone else because they have higher thoughts than we do and they express them better than we do.
Actually, some poetry lovers among the ordinary rank and file of the common speakers among the public have been heard to express that it is the emotion of seeing a universal worry/anxiety/reflection expressed in blindingly simple terms that really shocks them and gets them to identify with a poet and his message instantly. They say,for them,that is what makes a great poem - reading it, being gripped by it and then thinking 'Yes! That's how it is! That's exactly how it is for me!' For an example, see Seamus Heaney's 'Mid-Term Break' or Thomas Hardy's 'The Man He Killed' as many people might say that definition does not fit those poems. Of course,complicatedly, being able to express yourself in beautifully simple terms is an 'elevated skill' in itself!
The essence of what the quote is saying is that poetry has two elements:
(1) Poetry has thought.
(2) Poetry is expressed in language.
The commonality between the two is that both thought and language are in some ways elevated. I think we would need more context to determine exactly what "elevated" means. But just from the quote, it seems that the author is saying that poetry is intellectually superior and it is expressed in a superior style. In other words, poetry has form and content that is better.
'Thought' refers to the 'content'/ 'matter', and 'language' refers to the 'form' / 'manner'. The definition you have cited significantly contains the word 'elevated' as a common qualifier for both 'thought' and 'language' in poetry.
Something 'elevated' means that it is higher than the ordinary or commonplace, something lofty/noble/rarefied/transcending. If poetry is 'elevated thought', it must contain ideas which aim to transcend the limits of our mundane life, our lived existence. In that sense, poetry liberates us from the petty trivialities of life; it sublimates; it charts out a journey beyond, into the world of imagination, beauty, truth & joy.If poetry is an expression in 'elevated language', it means that the language of poetry must be somewhat vertically distant, essentially paradigmatic, to use the phrasing of Mallarme, 'to purify the dialect of the tribe'.
This definition of poetry is characteristically classical, may be very true of Milton's poetry, reminiscent of Longinus's notion of the 'Sublime'.