It's an extremely dangerous sport where one small slip can lead to serious injury.
Literal
Slightly make-mistake [if-と] big-injury [to-に] connect very dangerous [attributive-な] sport is-[polite-です].
A stern warning. The ~と conditional marks an automatic consequence — 'whenever X, Y follows.' 大怪我 ('serious injury') uses the 大~ intensifier prefix (for size/seriousness). 繋がる ('to connect, to lead to') is the standard verb for causal chains. 少し間違うと 'if you make even a small mistake' — the adverb 少し intensifies the smallness of the trigger.