What do you mean by "big races"?
Lets say Joe Blogs, aged 20ish rides with a local club.....how does he get noticed?
You start with local/regional races and then national ones. If he's in a local club then it's the local/regional races. As they move up categories - 4th>3rd, 3rd>2nd, 2nd>1st, 1st>Elite then access to 'big races' starts to open up; i.e. access to E/1/2 races rather than 3/4 or 4th only. British Cycling's 'Event' list is a useful starting point, filtering by the rider category, although others such as TLi exist. Note that it's not just road/crit races, but cyclo-cross, track, grasstrack, MTB, cycle speedway, etc. And time trials through Cycling Time Trials.
So they start as a Cat 4 racer, get used to racing in the 4th's or 3/4 races and develop racing skills. That
might get them noticed by a low-level team or they race under their club. Getting results gets them points - which allows a rider to move up a category. Once they're into 2/3/4 races then it's more likely that a rider is part of a team for Cat 2 riders and above, but not always. Some of the bigger events give priority to a team entry, with individual club/team riders getting the spare or reserve places.
So son no. 2's got 35 points as a British Cycling Cat 3 rider and needs 5 more to become Cat 2. To date this year I've got 15 points, which moved me from Cat 4 to Cat 3 once I'd got 12.
Keep going up the categories and bigger possibilities open up ...