Kevin's answer about core being used to hold the position is correct, but I wanted to add some clarifications about factors that can affect how you engage your core muscle when in an aggressive position:
- Total power output
- Hip position
- Overall flexibility (e.g., hamstring).
1) Power output
When you ride in an aggressive race position, you are usually at a high nominal power output. You will find that when pushing hard on the pedals, you naturally want to lower your upper body to counteract the force (Newton's third law). The best way I can describe it is that you are falling forward, but your pedal strokes keep propping your body up. When everything is in balance you will feel like you are barely touching the handlebars, enough to maintain control only. You need your core muscles to resist the forces, but its only basic core strength, nothing too crazy. Notice how bent the elbows of the riders are, you cannot support much weight for long periods of time with your arms bent.
If you are riding slower endurance paces (less total power output), you will need to shorten the reach and raise the stack as there is not enough pedal force to support your upper body in such an aggressive position. Here you will notice a lot of weight on your hands if your stack is too low. You can't look pro and go slow too without running into fit issues. The low stack height only works as a performance fit.
2) Hip Position
I know most do not think much about hip position when riding an aggressive position, but you need good hip flexibility so you can roll your hips forward to flatten out your back (blue rider pictured below is a good example). This makes it easier to engage your core muscles better to support your upper body. Try sitting in a chair, bending forward a bit (while keeping your feet off the ground) and see how easy or hard it is to support your upper body when rolling the top of your hips forward or backwards.
This passage from "Bike Fit" by Phil Burt (lead physiotherapist for British Cycling and a consultant physio for Team Sky) describes pelvis position.
Posture is the maintenance of a certain body position and requires appropriate joint mobility, joint/muscle coordination and muscular endurance. Limits in any of these elements can result in postural irregularities. Good posture on the bike required good flexibility through the hamstrings and the glutei muscles: this allows the pelvis to roll forward, keeping the back in a straight position while reaching the handlebars.
One major factor limiting the back's ability to remain relatively straight while on a bicycle is thoracic immobility: lack of movement in the middle of the spine normally results in the spine flexing too much. Excessive spinal flexion while on a bicycle will limit breathing and compromise your ability to stability your spine for torque production to the pedals
Note that not all the pros do this as young healthy bodies can get a way with bad position. For example, some rotate the top of their hips towards the back of the bike. This can cause some extreme spinal bends which could lead to back pain down the line.
3) General Flexibility
If you lack good general flexibility this position will be hard even if you can do 1) and 2). For example, if you lack hamstring flexibility you may not be able to get your hips into the proper position. In this case you may need to lower your saddle a bit (losing power) to make up for the lack of flexibility.