The mitzvah of shiluach haken is rooted in the pasuk commanding one who finds a nest to send away the mother bird before taking the young or eggs (Devarim 22:6), a mitzvah detailed further by Shulchan Arukh, Yoreh De'ah 292:1-2 and Tur, Yoreh De'ah 292 as applying only to a kosher, unowned bird.
The Gemara in Chullin 141a:5 and Bava Metzia 31a:15 discusses the repeated language of the mitzvah, requiring repeated dispatch of the mother if she returns, while Chullin 140a:14 and Chullin 138b:3 work out the scope of the obligation regarding kosher versus non-kosher birds and consecrated versus non-consecrated ones.
Kiddushin 40a:7 links shiluach haken to the reward of long life promised for the mitzvah, situating it among the Torah's rewarded commandments.
On the specific question of a brachah, Shaleach Teshalach, Halakhic Part, Responsa 4:9 explains, following the Rashba's principle that one does not recite a blessing over a mitzvah not entirely within the performer's control, that no blessing is made on shiluach haken in any manner—whether one first attempts to send the mother (where she might fly off on her own before the act is done, rendering the blessing in vain) or first takes her (where the mitzvah then comes through a form of transgression).
By contrast, Teshuvos V'hanhagos (Teshuvos V'hanhagos 3:345) considers whether one should recite a blessing before performing shiluach haken, suggesting that those who generally follow the Gra's approach to blessings might seemingly do so, though he leaves the matter uncertain.
Arukh HaShulchan, Yoreh De'ah 292:4-5 addresses the deeper rationale and scope of the mitzvah of taking the young, without addressing the question of a blessing directly.
Avnei Yashfeh 3:98 concludes that when the mother bird is found resting on a nest located on one's own windowsill or elsewhere on one's property, the mitzvah of shiluach haken does not apply at all, since the property automatically acquires the nest for the owner, rendering it 'mizuman' and exempt.