To understand what Matthew Arnold meant by the following lines from the poem "To Marguerite - Continued", we need to look at vocabulary, syntax, and context.
The vocabulary that might be unfamiliar to you consists of two words, "channels" and "sounds" used in marine parlance to refer to bodies of water. A "channel" in this context means a navigable body of water between land masses and a "sound" means a bay or an inlet.
The syntax is inverted, metri causa. The subject of
"And lovely notes, from shore to shore,
Across the sounds and channels pour"
is "notes" and the main verb is "pour". Thus, using a liquid metaphor, the notes pour, or travel across the water, to be heard by someone on the other shore. In context, this is analogous to (in the poet's imagination) the words of the poem written in England being read in France by the poet's beloved..