I studied the classical Libor market model, where the dynamics of rate $F_k$ from time $T_{k-1}$ to $T_k$ are given by $$ dF_k(t)/F_k(t) = \sigma_k(t) dZ_k(t) $$ under the forward measures $Q^k$ (where we use $P_k(t)$, the bond that matures at $T_k$, as numeraire). Then, it follows by a change of measure approach that for $i=k-1$, the dynamics under this measure $Q^k$ are $$ dF_{k-1}(t)/F_{k-1}(t) = \sigma_{k-1}(t) \left(dZ_{k-1}(t) - \frac{\rho_{k,k-1} \sigma_k(t) F_k(t)}{1+\tau_k F_k(t)} dt \right). $$ The above formula are taken from the book of Brigo and Mercurio and there the general formulation for a rate $F_i$ with $i<k$, resp., $i>k$ can be found.

I was able to technically follow the derivation of the drift, but what is the intuitive understanding of it? Can we use a trading argument to understand this equation?

This might be quite basic but I did not find such an intuitive explanation in the literature. Any comment or reference is highly appreciated.