What does let freedom ring mean?

The expression is from last line of the first verse of Samuel Francis Smith's poem "My Country 'Tis of Thee" (also called "America, the Beautiful"). The song itself is about America's people singing in celebration of their country --both its physical features and all the freedoms it enjoys (as the "sweet land of liberty"). The poet imagines the song being heard everywhere ('from every mountainside'), and even nature itself joining in the song.

The third verse continues and extends the image, looking for all people, all creatures, even the very rocks to sing out:

Let music swell the breeze,

And ring from all the trees

Sweet freedom's song;

Let mortal tongues awake;

Let all that breathe partake;

Let rocks their silence break,

The sound prolong.

Thus "let freedom ring" is a wish and invitation to fellow citizens that the "song of freedom" might be sung out throughout the whole land. The 'singing of the song of freedom' includes not only extolling the (past) enjoyment of liberty in this land, but also making it more and more a reality.