Why does the Star Trek franchise (produced in the USA) use Celsius for temperature and other units from SI, rather than Fahrenheit and units from the imperial system (still widely used in USA)? Eventually, as per Paul D. Waite's comment, the question can be, why don’t they use Kelvins?
Today, countries making up about 95% of the world's population use the metric system:
The holdouts are the USA, Liberia, and Myanmar.
If the Earth is peacefully united and sends missions to the stars -- as is the case in Star Trek -- the overwhelming majority of people would be metric users. Simple democracy would lead to the metric system being adopted.
Short answer? People feel that the future is the metric system. It's more endorsed by the scientific community. Many nations have adopted it as a universal measure. Thus, in a farflung science-heavy future, the assumption is that people will be using metric units exclusively, the same reason futurists thought people would all be speaking Esperanto in the future as seen in the Harry Harrison books.
You answered your question in the question - because Celsius is an SI unit (well it's not really, Kelvin is, but Celsius is just a constant offset so it is for the purposes of this question). There's no logic to a scientific organization in the future using anything other than what the scientific community use (Nasa used imperial for a while because it was US based).
Other people have mentioned the fact most of the world use Celsius, but this is irrelevant. While it's sensible for countries to use SI, even if no one used Celsius, it would still be adopted by any scientific organization. An example is acceleration, where no country (as far as I know) would quote acceleration in m/s^2 but that's what science uses.
The Federation is a utopian society derived from Earth. Such a utopian future world would use a consistent and planned measurement system. Thus they use Celsius. Because it is logical and simple. One Celsius degree is the same as one Kelvin which is 1/100th of the total range from the freezing point to the boiling point of water (at 1 atmosphere pressure). Fahrenheit is much more complicated scale (see here).
Kelvin are much more unwieldy at the temperatures that we are accustomed to. Warm summer day is 298K or 25 degrees C. Note, the size of the Kelvin is set as the same as the degree Celsius. Kelvin just starts at 'absolute zero' which is -273.15 deg C.
So Celsius was probably chosen because it is consistent, logical, simple, yet relatable by average audience (Americans in the Sixties), and it was also 'futuristic' to non-scientists at the time ST was invented.
Here and there both systems are used - sometimes just I think because 'miles' and 'inches' are easier to grasp in the mind and feel more human. However look how metricated the whole mythos is at its core - from stardates to coordinates. Therefore it is very logical they should use Celcius as well.
Most importantly of all; Trek represents a utopian future where mankind has joined together without negative nationalism nor bigoted jingoism. In this single culture the sheer number of humans who do measure things in tens would massively outweigh those who don't. Logic - and therefore the metric system - would prevail through democracy just as @Royal says. Metric is also the measurement system of Science and Trek is a high Technocracy.
Star Trek is just a story. SI is just another future ideal world. The real world in fact produces a lot of incompatible systems, hard to use computers, etc., and uses imperial units in lots of places. I am talking here about the so-called metric world, e.g. The Netherlands. My bicycle has 28 inch wheels. Electronics goes in 19 inch racks. Spacing of integrated circuit leads are in 1/10 inch. (Ok, 1/10 is kind of metric.) Resolution of printers and photos is in dpi dots per inch. Examples of non-imperial, but also non-metric units used here: light year, oil production in barrels.
I think the "universal translator" takes care of it, as does the specialized translators used for ships log entries etc.
If Spock were to use a cultural reference in his Officer's Log, and speak of "a hundred twenty eight squelm" in FedStandard (which is decendent from and rendered as English in the show) the metadata would automatically note the standard value in kelvin, and later when a sulfer-breathing admeral from Sarr reads it, it will be in his native language with the value in kelvin and a footnote explaining that the author likened it to the desert mesa whatever blooms are triggered, with links. Or, it may show a notation mapping to the normalized clement range of the author, so he knows without distraction if that is supposed to be hot or bitter cold or whatever. In the case of a human reading, Fahrenheit might be one of the configurable options of the normalized clemency perception scale.
Since Starfleet is primarily founded and organized by Terran and Vulcan world governments, whose to say SI is the end-all/be-all of measurements? They might use Vulcan-based Interplanetary Standard units.
Your question isn't entirely accurate. Star Trek uses imperial measurements. :)
In Star Trek, the original series, they use imperial. E.g. Spock tells Kirk a temperature in Fahrenheit, and at some point they both look at Mudd's data file and it gives his height in feet.
They also use metric, sometimes in the exact same episode for the same measurements (e.g. distance). It's a big mix.
From a production standpoint this is presumably because the writers at the time didn't put much thought in and just wrote what they know.
For scientific purposes metric is the accepted standard, so for those above the Enterprise (all of whom have some degree of scientific expertise) it would simply be natural. On top of that, astronomical units of measurement are based in the metric system for example, we use Km to measure near planetary distances, it's only when we get up to interplanetary distances that we start to measure in AU (the distance from the Earth to the Sun), but gigametres are also interchangeable here. As we continue increasing in size above parsecs we have the kiloparsec and megaparsec which are all metric units of measurement.
As for why they don't use Kelvin - Kelvin has a straight conversion of just being 0 Celsius + 273.15 so the two are interchangeable. We can assume that they use the two interchangeably as the scientific community does, so for ambient temperatures will refer to 25C rather than 298.15K but when referring to the very cold might use Kelvin. So they probably do use Kelvin but it's all based on context, for example I don't say that my walk to the kitchen is 0.003Km from my sitting room, instead I just say it's 3m.