Allusion is a literary technique using specific language to refer to something else of significance, either directly or indirectly. In the phrase indicated, the narrator is using religious language to speak about the feeling he is getting from the thrush's song. The phrase "Blessed Hope" is from the Epistle to Titus, attributed to St. Paul. The phrase in question is:

...while we wait for the blessed hope -- the appearing of the glory of our great God and Savior...
(bible.cc)

In this context, the Blessed Hope is the Second Coming of Jesus, which is prophesied in most versions of the New Testament. Using this information, as well as the previous religious imagery in the poem, the phrase used here is a good example of allusion, as it alludes to the Second Coming through the beautiful birdsong in the middle of winter.

So little cause for carolings
     Of such ecstatic sound
Was written on terrestrial things
     Afar or nigh around,
That I could think there trembled through
     His happy good-night air
Some blessed Hope, whereof he knew
     And I was unaware.
(Hardy, "The Darkling Thrush," poets.org)

Previously, the narrator referred to the thrush's song as "evensong," which is an Anglican evening prayer. This makes the allusion to the Second Coming easier to see, and more effective as a metaphor: just as the thrush brightens the gloomy night with its song, the Second Coming is said to herald a new age of peace and enlightenment. This adds another layer to the emotional feeling of the narrator; not only does the thrush make the foreboding winter seem more bearable, but it reminds him that there is hope for the rest of the world as well, hope which extends beyond his own simple life and into the realm of the spiritual.