So there's three parts to this question: 1) Fast copying of objects, 2) Lining up row, 3) making these objects show appropriately spaced 'chunks' of an image. The good news is you can do the whole thing in about 20 seconds.
1) The fastest way to copy loads of the same thing in illustrator is: hold down alt, press an arrow key, and it creates and selects a copy nudged one unit in that direction. (it's alt on Windows, on a Mac it might be alt or maybe option?) Hit alt-rightarrow 10 times and you'll get 10 copies all perfectly vertically aligned, almost on top of each other (holding shift as well gives more space)
2) To line them all up evenly in a row, a fast easy way to space them perfectly is to drag the far-most one to where you want the row to end, select all of them, open the awesome and underrated 'Align' window (Window > Align), and hit the 'Horizontal distribute space' button.
You can easily adjust the amount of spacing using this method: simply adjust the position of the far object, and/or delete or copy objects from the middle, and/or resize all, select the row and hit the button again, and it'll space them perfectly for you again. Repeat until you've got it looking just how you want it right.
When it's how you like it, you can then select this whole row, group it, and do the same thing vertically: hit alt-uparrow 3 times, drag the top row to the right place, select the rows, hit 'Vertical distribute space', tweak until it looks right, then select all four groups, ungroup, and delete those objects you don't need.
This is probably the fastest way [edit: second fastest, after plainclothes' excellent technique] - the whole task can take a few seconds, including tweaks and adjustments.
There are other ways too. It's almost as fast to create then expand a fixed-distance or fixed-number blend between two objects horizontally, then again between two groups vertically: doing so also gives you the option of having something change or fade gradually between each object, like colour, opacity or shape. That's a seperate topic, but here's a neat example of the basic principle (plus other stuff) if you're interested.
3) To easily get the effect in the linked image, stick your grid of objects on top of a placed image. Select them all and make them into a compound path (object menu, or ctrl-8). Select this compound path and the image, underneath, and create a clipping mask (object menu, or ctrl-7). Magic!
(I included some expanded text here as well, purely because it looks cool. Note that you need to expand text into fixed paths (object menu) and include it in the composite path in order to use it in a clipping mask like this)