If you want the background of a parent control you can do like this:
<Window x:Class="GlowingTextBox.MainWindow"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
xmlns:local="clr-namespace:GlowingTextBox"
mc:Ignorable="d"
Title="MainWindow" Height="350" Width="525">
<StackPanel Background="WhiteSmoke"> <!-- The parent with the background color you want for your text box -->
<TextBox Width="200" VerticalAlignment="Center" HorizontalAlignment="Center" Name="MyText">
<TextBox.Style>
<Style TargetType="{x:Type TextBox}">
<Setter Property="Margin" Value="20" />
<!-- Bind the bacground to the stackpanel -->
<Setter Property="Background" Value="{Binding RelativeSource={RelativeSource AncestorType=StackPanel}, Path=Background}" />
<Setter Property="Effect">
<Setter.Value>
<DropShadowEffect ShadowDepth="0" Color="Yellow" Opacity="0" BlurRadius="20"/>
</Setter.Value>
</Setter>
<Style.Triggers>
<Trigger Property="IsFocused" Value="True">
<Trigger.EnterActions>
<BeginStoryboard>
<Storyboard>
<DoubleAnimation To="1.0" Storyboard.TargetProperty="(Effect).Opacity" Duration="00:00:00"/>
</Storyboard>
</BeginStoryboard>
</Trigger.EnterActions>
<Trigger.ExitActions>
<BeginStoryboard>
<Storyboard>
<DoubleAnimation To="0.0" Storyboard.TargetProperty="(Effect).Opacity" Duration="00:00:02"/>
</Storyboard>
</BeginStoryboard>
</Trigger.ExitActions>
</Trigger>
</Style.Triggers>
</Style>
</TextBox.Style>
</TextBox>
<TextBox />
</StackPanel>
</Window>
EDIT: In the above I've set the background of the textbox to reference the background of the stackpanel that is parent to the textbox. It seems that when you use dropshadoweffect it shines through the textbox if it has baground set to transparent. So if you want the textbox to have the same background as its parent (here stackpanel) you references it as <Setter Property="Background" Value="{Binding RelativeSource={RelativeSource AncestorType=StackPanel}, Path=Background}" />
If you want the textbox to have its own background you do this:
<Window x:Class="GlowingTextBox.MainWindow"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
xmlns:local="clr-namespace:GlowingTextBox"
mc:Ignorable="d"
Title="MainWindow" Height="350" Width="525">
<StackPanel Background="WhiteSmoke">
<TextBox Width="200" VerticalAlignment="Center" HorizontalAlignment="Center" Name="MyText">
<TextBox.Style>
<Style TargetType="{x:Type TextBox}">
<Setter Property="Margin" Value="20" />
<Setter Property="Background" Value="White" />
<Setter Property="Effect">
<Setter.Value>
<DropShadowEffect ShadowDepth="0" Color="Yellow" Opacity="0" BlurRadius="20"/>
</Setter.Value>
</Setter>
<Style.Triggers>
<Trigger Property="IsFocused" Value="True">
<Trigger.EnterActions>
<BeginStoryboard>
<Storyboard>
<DoubleAnimation To="1.0" Storyboard.TargetProperty="(Effect).Opacity" Duration="00:00:00"/>
</Storyboard>
</BeginStoryboard>
</Trigger.EnterActions>
<Trigger.ExitActions>
<BeginStoryboard>
<Storyboard>
<DoubleAnimation To="0.0" Storyboard.TargetProperty="(Effect).Opacity" Duration="00:00:02"/>
</Storyboard>
</BeginStoryboard>
</Trigger.ExitActions>
</Trigger>
</Style.Triggers>
</Style>
</TextBox.Style>
</TextBox>
<TextBox />
</StackPanel>
</Window>
EDIT: Here the background of the textbox is set to white in the style to let it be indifferent of underlying colors.