In the context of driving, the verb ease describes slow, careful movement, and the term eased out describes slowly merging out of some place, while eased out on (or eased out into) can mean merging onto a busy street, or into traffic. The person doing the "easing out" is generally the driver.
It's not clear to me if your cousin was giving you advice to help you off the narrow street, or if he actually got in the driver's seat to get you out of a tricky driving situation.
If your cousin was driving, here's a sentence you could use:
I'm new to driving a car, so my cousin had to ease us out so we could get through the narrow street.
If he was just giving advice, you might say:
I'm new to driving a car, so my cousin helped me ease out into the narrow street.
Here are a few more examples; hopefully, they can help you better understand how to use the verb ease in relation to driving:
The driver eased the car through the narrow street, being careful not to hit the pedestrians.
When we use ease out, it means the car is going "out" somewhere, like out of a parking spot, or out onto a highway:
The driver of the silver car is easing out into the rush hour traffic.
We can also ease into instead of ease out on:
The silver Fiat eased into a tight space between a minivan and a pickup truck.