Relative deprivation theory proposes that social groups (race, class, ethnicity, religion, sexual orientation, etc.) organize social movements to advocate for obtaining resources (e.g., rights, status, political representation, equality, money) in which they feel they are deprived relative to other social groups; this deprivation of resources can be either actual or perceived.
Applying this theory to the Black Lives Matter movement, one might start with the movement's founding. BLM was created in 2013 in response to the death of Trayvon Martin at the hands of George Zimmerman. BLM's mission, as stated on their website, is to "eradicate white supremacy and build local power to intervene in violence inflicted on Black communities by the state and vigilantes" (https://blacklivesmatter.com/about/).
A relative deprivation theorist might argue that BLM was created to call attention to the deprivation of Black Americans' civil rights and liberties, which occurs through the disproportionate level of state and vigilante violence leveled against Black Americans. A RD theorist might also argue that Black Americans suffer from discrimination, poverty, undesirable healthcare outcomes, and poor political representation relative to other racial groups, all of which BLM calls attention to and works for political and social resolutions. Through a RD lens, Black Americans are unfairly deprived of resources that other groups have, which leads to the rise of social movements like BLM to correct such injustices.