The reaction mechanism was first investigated by Scott Searles and coworkers at the University of Missouri.  Overall, the reaction can be thought of as a reductive coupling of the carbonyl compound and the terminal alkyne.  In the Crabbé reaction, the secondary amine serves as the hydride donor, which results in the formation of the corresponding imine as the byproduct.  Thus, remarkably, the secondary amine serves as Brønsted base, ligand for the metal ion, iminium-forming carbonyl activator, and the aforementioned two-electron reductant in the same reaction.  
In broad strokes, the mechanism of the reaction is believed to first involve a Mannich-like addition of the  species into the iminium ion formed by condensation of the aldehyde and the secondary amine.  This first part of the process is a so-called A coupling reaction (A stands for aldehyde-alkyne-amine).  In the second part, the α-amino alkyne then undergoes a formal retro-imino-ene reaction, an internal redox process, to deliver the desired allene and an imine as the oxidized byproduct of the secondary amine.  These overall steps are supported by deuterium labeling and kinetic isotope effect studies.   Density functional theory computations were performed to better understand the second part of the reaction.  These computations indicate that the uncatalyzed process (either a concerted but highly asynchronous process or a stepwise process with a fleeting intermediate) involves a prohibitively high-energy barrier. The metal-catalyzed reaction, on the other hand, is energetically reasonable and probably occurs via a stepwise hydride transfer to the alkyne followed by C–N bond scission in a process similar to those proposed for formal [3,3]-sigmatropic rearrangements and hydride transfer reactions catalyzed by gold(I) complexes.  A generic mechanism showing the main features of the reaction (under Crabbé's original conditions) is given below:(The copper catalyst is shown simply as "CuBr" or "Cu", omitting any additional amine or halide ligands or the possibility of dinuclear interactions with other copper atoms.  Condensation of formaldehyde and diisopropylamine to form the iminium ion and steps involving complexation and decomplexation of Cu are also omitted here for brevity.)
Since 2012, Ma has reported several catalytic enantioselective versions of the Crabbé reaction in which chiral PINAP (aza-BINAP) based ligands for copper are employed.  The stepwise application of copper and zinc catalysis was required: the copper promotes the Mannich-type condensation, while subsequent one-step addition of zinc iodide catalyzes the imino-retro-ene reaction.