First, make a list (by inspection of the Navier-Stokes equations) of what variables the entrance length $L_{\textrm{e}}$ might depend on. One comes up with fluid density $\rho$, dynamic viscosity $\eta$, (mean) velocity $u$, and pipe diameter $D$. Hence there are five variables altogether. Those variables contain three base dimensions (mass, length, and time). Hence, whatever the relationship is between them, the Buckingham $\pi$ theorem tells us it can be expressed as a relationship between $5-3 = 2$ dimensionless variables; that is to say, if we can construct two distinct dimensionless groupings of the variables, then the value of one of those groupings depends only on the value of the other. We construct dimensionless groupings $L_{\textrm{e}}/D$ and $\rho u D/\eta$ (the second one being the Reynolds number). We know those groupings are distinct, because the first one has no involvement of $u$ and the second one has no involvement of $L_{\textrm{e}}$ (the "non-repeating variables"). Hence, we know that $L_{\textrm{e}}/D$ depends only on the Reynolds number. An algebraic way of writing that statement is $L_{\textrm{e}}/D = f\left(\mathit{Re}\right)$, or equivalently $L_{\textrm{e}} = Df\left(\mathit{Re}\right)$ (actually a stronger statement than the one you were looking to prove).
