This page of the Cyberpunk 2077 guide explains how to unlock car racing in Night City. You will find out how to get access to races and what are their rules.
Street races are initially not visible on the city map. This is not a bug, because you will only be able investigate this topic more as part of related side missions.
You must wait until Claire Russell calls V. You will find out from her that she needs a driver for street racing. If you express your interest in her proposal, a related side mission - The Beast In Me - will appear in your journal.
You must visit Claire's garage located in the Arroyo district of Santo Domingo. Talk to Claire and she will explain the general rules of car racing in Night City - one person drives the car and the passenger tries to attack the competitors.
You cannot start the first races immediately. You have to wait until Claire calls V again. The races take place in the City Center, Badlands, Santo Domingo and Watson - get to those districts and wait for Claire to contact you. You can then respond to her message.
Each race is a separate entry - a side job in your journal. Represented as yellow icons on the map.
Talk to Claire and decide whether you want to use your own vehicle in the race or Claire's (Beast). In the initial stage of the game, the latter is better because its a massive pick-up that's very durable and at the same time has acceptable performance stats.
In order to start the race you have to sit in the driver's seat of the vehicle you will use in the race. Claire will sit in the passenger seat.
During the race, focus solely on flawless driving - arrows indicating the route are displayed on the track. You must correctly drive through all control gates.
Listen to Claire's comments, as some of them will be related to upcoming acute turns. Claire can do all the shooting without any involvement on your part.
Winning the race is rewarded with credits , and in addition to passing each of them you have to be on the podium, i.e. finish the competition in the top three.