Physical interpretation of the change of diffusion term in navier stokes equations In the Navier-Stokes Equations, there is one term accounting for convective flow and one term for diffusive flow. At high flow rates, the diffusive term becomes much smaller compared to convective term, therefore can be neglected, leading to Euler's equations. 
I can understand this in terms of mathematics: There is a multiplicative factor $\frac{1}{\text{Re}}$ in the diffusive term. With increasing flow rates, the inertial forces become larger than the viscous forces, so the  Reynolds number increases and the term for diffusive flow becomes much smaller. But what is the physical interpretation of this? Does the influence of random motions decrease when the flow rate increases? 
 A: It's not that the random motion decreases when the flow rate increases. It is only that the random motion stays the same but the coherent motion dominates. If the diffusion velocity in a gas is 1 and the convective velocity of the flow is 1000 (units don't matter), then the diffusive action can be pretty safely ignored. 
The important thing to remember is that there are limits to where the approximations can be applied. At high Reynolds number then one can use the Euler equations ignoring viscosity outside of the thin region around bodies where no matter how large the convective velocity is there will always be viscous effects there.
A: You almost give the answer in your question: 

With increasing flow rates, the inertial forces become larger than the viscous forces

We can state it more precisely: The inertial forces scale quadratically with the flow speed $U$: $\rho \mathbf u\nabla \mathbf u\sim \rho U^2/L$, while the viscous forces scale linearly: $\mu \nabla^2\mathbf u \sim \mu U/L^2$. Here $\rho$ is fluid density, $\mu$ dynamic viscosity and $L$ is a characteristic length scale.
The ratio between the two is the Reynolds number $\mathrm{Re} = \rho UL/\mu$. Therefore, when $\mathrm{Re}$ is large, the viscous term may be treated as small.
Now, I cannot downvote, but I'd like to amend a misconception present in the question and the two existing answers.
The viscous term $\mu \nabla^2\mathbf u$ is not to be interpreted as "diffusive flow", nor as molecular diffusion (as described by the Péclet number.) Instead it describes the effects of internal friction in the fluid. Friction arises when neighboring fluid parcels have different velocities. The effect of friction is to even out the velocities, thus reducing flow velocity gradients. In fact, the very form of the viscous term comes from the assumption of a Newtonian fluid: the friction force is proportional to the local flow velocity gradient$^1$, and the constant of proportionality is the dynamic viscosity $\mu$.
It is common practice to call the viscous term "diffusion", presumably because of the second derivative. The quantity "diffusing" in Navier-Stokes is the fluid velocity, in the sense that high fluid velocities diffuse toward regions of lower fluid velocities, in the negative gradient direction. But in contrast to molecular diffusion there is no randomness involved in the viscous term$^2$. The irreversibility is a result of frictional dissipation.
I'll re-use an important paragraph from tpg2114's answer:

The important thing to remember is that there are limits to where the
  approximations can be applied. At high Reynolds number then one can
  use the Euler equations ignoring viscosity outside of the thin region
  around bodies where no matter how large the convective velocity is
  there will always be viscous effects there.

We now understand why this is: the viscous term actually scales with fluid velocity gradients, while the convective term scales with the fluid velocity itself. Therefore, near a stationary boundary where the velocity is zero, it follows that the velocity is small and the gradients are large, hence the value of $\mathrm{Re}$ is locally small, and there's a boundary layer.
Finally, I'd like to comment, also with respect to tpg2114's answer, that units do indeed matter. The important thing for the inviscid approximation to hold is that the dimensionless Reynolds number is large. Even if we among friends may say "for large velocities..", we must understand that we mean "for large values of the Reynolds number" in this case. It is similar to quantum mechanics where we may say "because $\hbar$ is so small..", but in fact we usually employ units where $\hbar=1$.

$^1$ The particular form for a Newtonian fluid arises from the statement that the stress $\tau_{ij}$ (force per area due to pressure and friction) is given by $\tau_{ij} = -p\delta_{ij} + \mu (\partial_ju_i+\partial_iu_j)$. The Navier-Stokes equations are in fact momentum conservation equations stating that the divergence of the stress tensor equals the time rate of change of momentum:
$$
\frac{D\rho u_i}{Dt}=\partial_j\tau_{ij}
$$
$^2$ However, it must be said that the friction (viscosity) is of course a consequence of random collisions on the molecular level of modeling. The viscosity often has a strong dependence upon temperature, for example. But on the continuum level, this randomness is averaged into macroscopic quantities and modeled by constitutive equations like the Newtonian fluid above.
A: It is helpful to think of this problem in terms of timescales.
Look at the navier stokes equation in 1D:
$\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = - \frac{1}{\rho} \frac{\partial p}{\partial x} + \nu \frac{\partial^2 u}{\partial x^2}$
Considering timescales on the order of diffusion for a domain of length, $L$, the diffusion timescale is, $\tau_d \sim L^2/\nu$. On the other hand, the convective timescale, where $U$ is a typical velocity in the flow, is $\tau_c \sim L/U$. In other words, the timescales of each process are disparate by about $Pe = \tau_d/\tau_c \approx UL/\nu$ times. The latter quantity is known as the Peclet number, $Pe$, which compares the relative size of each time-scale. When $\tau_c << \tau_d$ or $Pe >> 1$ then transport is dominated by advection/convection. If $Pe << 1$ diffusion dominates.
You can plug in different values and see verify for yourself that only if $Pe$ is close to 1 are the diffusive and convective time-scales on the same order. In typical flows where the timescale of diffusion is much larger than the timescale of convection, the Euler equations can be employed to a good approximation.
