Because Tacitus, who is our main source on her revolt, describes her as riding around and exhorting troops from her chariot. From the Annals, Book 14:

Boudicca, mounted in a chariot with her daughters before her, rode up to clan after clan and delivered her protest

You've gotten lots of answers about the Britons and their chariots and Boudica and her chariot. I think it's worth looking at your belief that the Celts were not skilled at riding horses, because the Gauls, Britons, and Galatians were all pretty famous for their skill with horses and wheeled vehicles. Indeed, most of Latin's words for wheeled vehicles aren't Italic in origin, they're loan words from Gallic. The Galatians were pretty much all mounted in carts and chariots when they invaded, and Caesar is quite praiseworthy of Gallic cavalry, which kicked his own cavalry's ass pretty thoroughly until he got his hands on German mercenaries, who were aggressive enough to challenge and rout the Gauls.

hi OP! you might be interested in some previous posts on chariots in Britain

• How did chariot battle tactics differ between the middle-eastern nations (say, at the height of the Persian Empire) and those of the British isles (say, during the Roman invasion)? How were they similar?

• From about 400 BC to about 100 BC, what did people find scarier to face, a cavalry charge or a chariot charge?

and also general information about the use of chariots in warfare and subsequent decline. Basically, it's important to know that European horses used to be too small to carry people, hence they were used to pull carts like chariots. So it's worth taking a look through these threads to get an idea of whether horses would be large enough to ride in Britain during Boudica's time

• Why did the ancient Mesopotamians use chariots in lieu of cavalry for so long? Why did this change? <-- a bunch o' links; check out the ones under 'decline'

• a more recent post Why was the use of chariots in wars popular in antiquity but not in the middle ages?