。
The front wheel plays a major role in keeping a two-wheeled vehicle running upright without falling over.
Literal
Two-wheeled-vehicle [subject-が] not-falling [without-ずに] running [nominalizer-の] [for-には], front-wheel [subject-が] big role [object-を] is-playing.
Two grammar points worth tagging. (1) ~ずに ('without doing X') is the formal/written negative connective, equivalent to ~ないで but more elevated in register — it attaches to verb stems via the negative ず. So 倒れずに = 'without falling.' (2) のには is a composite particle: の nominalizes the preceding clause ('the running of the bicycle'), に marks it as the goal/purpose, and は contrasts that purpose with whatever else is on the table — 'as for [the purpose of] X-ing.' The whole frame [verb] のには [agent]が[role]を演じる is the standard 'play a role in [doing X]' construction.