How does vasodilation slow blood flow?
The measurement that decreases during vasodilation is flow velocity, $v$, or a change in distance over change in time $\frac{\Delta x}{\Delta t}$. Prior to a reflex response, the same rate of flow, $Q$, or $\frac{\Delta V}{\Delta t}$ enters the capillary bed (here we use a capital V to represent volume, where $\Delta V$ is a change in volume, and $\frac{\Delta V}{\Delta t}$ is the change in volume over change in time. Because volume is area times a third distance dimension (e.g., if we say area is height times width, then volume is area times depth or thickness), we can describe flow as flow velocity times cross sectional area ($Q = A \cdot v$, or  $Q = A \cdot \frac{\Delta x}{\Delta t}$). If you increase the cross sectional area, $A$, and maintain the same flow, $Q$, then flow velocity, $v$, has to decrease.

A more intuitive explanation would involve turning on a garden hose and covering part of the end with your thumb. The smaller the opening you allow, the greater the flow velocity through the end of the garden hose. The same volume of water leaves the hose per unit time. It's just moving at a faster flow velocity if you decrease the cross sectional area (and at a slower flow velocity if you increase the cross sectional area). 

This is discussed in Chapter 4 of Costanzo Physiology, in the section on hemodynamics.

The actual physical measurements end up being more difficult to predict in a situation with parallel capillary beds and reflex responses. $Q$ does not stay constant, for both the specific capillary bed, and for the entire circulatory system. However, you can say that, as a general rule flow velocity, or $v$, will decrease when cross sectional area, $A$, increases. As @BryanKrause points out in particular, provided the series resistance for a capillary bed doesn't change, the initial $Q$ for the capillary bed won't change, which means this caveat isn't relevant.