"quickly" means that the action takes/is over withing a short amount of time. "fast" refers to the speed with which an action is performed. There is definitely some correlation since the faster you do something the quicker the action will be over. So it might depend on the specific context and what you want to stress. "Go quickly to the store and get me some milk, please." This does not mean that the person is asked to walk fast or even to run to the store, but that the action of going to the store will not take much time and they will be back with the milk within a short time.
Without any further context, "fast" is generally the implied meaning, particularly in combination with "running".
It's part of the consonant gradation system. In the same way that nainen > naisen in the geneitive, the s in yousta is a reduced form of the ks in the stem juokse-.
From Wikipedia (https://en.wikipedia.org/wiki/Finnish_verb_conjugation)
Nonderivable and irregular stems Standard Finnish has comparatively very few irregular verbs in addition to 'olla' discussed above. However, because the infinitive is an inflected form of the root, the consonant gradation may obscure the root. The root of the word 'juosta' = 'to run' is juoks-; when generating the infinitive, the pattern ks → s is applied: Template:Juoks+ta → juosta. Epenthetic 'e' is added for personal forms: juoksen.
There is a rare pattern with a stem with -k- rendered as -hdä in the infinitive but disappearing in gradation:
'tehdä' = 'to do, make': tee-; teen, teet, tekee, teemme, teette, tekevät, etc. 'nähdä' = 'to see': näe-; näen, näet, näkee, näemme, näette, näkevät, etc.
That is, teke- and näke- forms are rendered as tehdä and nähdä in the infinitive but are subject to gradation of 'k' in personal forms like teen. In some colloquial forms, the 'e' is rendered as a chroneme instead: nään instead of näen etc.
To get very technical about terminology...
From what I've seen on Wiktionary about juosta and other type 3 verbs, jousta might be descended from jooksetak. There was a sound change that elided 'e' between 's' and 't', which would have created jookstak. This is the same sound change that turned naiseta to naista.
The first consonant of a three consonant cluster was often then deleted, which would have created joostak. And sometime between Proto-Finnic and modern Finnish, word final 'k' was reduced to an unwritten glotal stop and 'oo' became 'uo', producing juosta.
But then juoksen would be descended from ancient jooksem, then from jooksen. If so, the 'e' in juosta is elided, rather than epenthetic in juoksen, diachronically at the very least.
And paraphrasing Wikipedia, consonant gradation is open syllables using the strong grade, and closed syllables using the weak grade. Consonant gradation was used in Proto-Finnic, when the infinitive was joostak and the 1st person singular was jooksen.
Both had closed syllables, which would require them both to be in the weak grade, if consonant gradation was in play. And in modern Finnish, consider closed syllable juoksen and open syllable juoksevat. So I'm not sure that consonant gradation is the best term for the juosta/juoksen consonant mutation.