It's really useful in cases where you need an answer a question like "I want to drill a hole in the middle of this board which is 12.25 (12 and 1/4) inches long." It's easier to for me to split that in half in my head than divide the fraction and find the corresponding mark on my tool.
Half of 12 is 6, half of 1/4 is 1/8, I drill the hole at 6 and 1/8".
It's easy to get a fractional measurement of whatever is in front of you and immediately work with that measurement in your head then find the corresponding mark on your measuring tool. There is no reason the same system would not work with metric, it just seems to be more common with imperial units.
In metric-land, standard measurements for things are used to make things like that easily divisible.
E.g. IKEA cabinets are 60 cm wide (90, or 120 for larger cabinets), common countertop depth is 60 cm, and so on.
That way we get to use sane and consistent units and we can easily see/measure the middle of a 60cm piece of wood or whatever.
It also tickles me how eyeballing these things is apparently something desirable. Whenever I work around my house I measure everything twice to be sure...
This has nothing to do with eyeballing, I hope I did not give that impression. In the imperial measurement world there are also standard sizes of things but that's not the point. Using a fractional scale (not necessarily imperial!) does have benefits in some situations, such as measuring and working with some arbitrary object placed in front of you. Measuring twice is not unique to the metric system and is very common in the imperial unit world as well.
Not everything in the world is (or should be) provided by Ikea :)
Half of 12 is 6, half of 1/4 is 1/8, I drill the hole at 6 and 1/8".
It's easy to get a fractional measurement of whatever is in front of you and immediately work with that measurement in your head then find the corresponding mark on your measuring tool. There is no reason the same system would not work with metric, it just seems to be more common with imperial units.