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The technical science behind Photoshop is not my strength, but here's my layman's understanding of how frequency separation (FS) works:

• In general, the purpose of FS is to extract and separate the texture from the color. The classic use of FS is for skin retouching, where you want to smooth-out blotchy or damaged skin while retaining the intricate pattern of pores, wrinkles and the non-damaged areas of skin texture.
• To achieve this, FS employs the Subtract blend mode (as a setting found within the command Image>Apply Image). The Subtract blend mode does a mathematical calculation—subtraction—of the brightness levels between the two layers that are being "applied" to each other. If you have two identical layers and subtract one from the other, the result will be pure black because you are subtracting a given level of brightness from itself, leaving "nothing" behind... zero brightness.
• In order for the Subtract blend mode to do anything useful, the two layers you are subtracting must have something different about them—different levels of brightness—so that there is something left behind that is non-black. This is where the blurring step in frequency separation becomes essential.
• Let's say we have this image below and we want to extract the sandy texture from the green background by using FS. We first create two identical layers. Then, in order for the Subtract blend mode to "identify" the sandy texture, we need to blur the bottom layer to make that texture different enough from the top layer so that it is distinguishable. If we now apply the Subtract blend mode (within the Apply Image command), the green areas subtract from themselves and leave nothing behind (i.e., black), but the grainy parts are now different enough—from one layer to the other—that the grainy pattern remains behind.
• In the Apply Image command, we must specifically set the "offset" box to the number 128. A brightness level of 128 is exactly 50% Gray. So now, instead of the Subtract blend mode resulting in black whenever the two layers are identical, it results in 50% Gray instead. We can then make 50% Gray disappear in the top "texture" layer by applying a blend mode of Linear Light. Other blend modes (Overlay, Hard Light, etc.) will also make 50% Gray invisible, but Linear Light is the recommended one. Once you've made all the 50% Gray disappear, you're left with just the pure texture pattern in your Linear Light top layer.
• Most FS tutorials I've seen don't tell you exactly how much to blur the bottom layer. What I've learned through trial & error is that you first identify the textures you want to retain via FS. Then, on the bottom layer, you blur just enough until that texture disappears or is smoothed away, but not so much blur that you introduce too much brightness differences in the "color" portions of the image that you're not interested in.

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Hope this makes sense and is not to wordy.