The transwarp conduits were largely guarded by the Borg, making them pretty dangerous in general.
Also, bear in mind that these conduits were features of space; even the Borg needed very special technology to create new conduits. So even if the Federation wanted to use the conduits and brave the Borg, they only knew where one or two conduits were as of TNG.
They do use the technology in other series, most notably Voyager. The problem is they don't really know how it works, so any time they use the technology, it tends to be a giant hack that barely works, if at all.
In the last episode of Voyager,
Captain Janeway had to travel through time to get information to the old crew on how to successfully navigate the Borg transwarp conduits.
In DS9, about the same time as the Voyager crew were experimenting in the Delta Quadrant, Starfleet was attempting to learn more about transwarp capabilites to help in the Dominion war, but they still hadn't made huge progress.
Even with former Borg helping, such as Seven-of-Nine and Captain Picard, the technology was just too far advanced for the Federation to master at the time.
In Star Trek Online, about 35 years after Voyager and DS9, players have access to transwarp drives as a fairly typical technology. However, I'm not sure if that's canon (well, as canon as an MMO gets), or just there as a gameplay device because spending three days crossing an imaginary galaxy would be boring.
Of note (more about Voyager's last episode),
future Janeway was from 2404, 5 years before STO, and had very recently perfected all the tech she needed.
So it stands to reason the Federation could have started using transwarp technology by the time STO takes place.