The solution to the Fermi Paradox is understanding exponential growth.
At a given speed, the volume of space that can be reached in time t growth in proportion to the cube of t. The population that needs to be hosted, or the resources that need extracting, however, grow exponentially with a given growth rate.
The thing is that the exponential growth catches up with the cubic growth real fast. So if you believe continued growth is a requisite for a healthy civilisation, colonising space won't help, the rot of stagnation and hell of extreme scarcity is going to catch up with us in no time, on astronomical scales. If, on the other hand, you believe we can wean ourselves off the necessity of having ever more people consume ever more resources and still maintain a thriving civilisation, space colonisation becomes unnecessary - we can do that right here on earth.
You'd think we all gained a bit of an understanding of exponential growth after two and a half years of the pandemic, but the speed with which this happens is still counterintuitive. For example, with a growth rate of just 1%/ year and an expansion speed of c, the c tipping point where demand for more space grows faster than the supply of new star systems is reached in a mere 301 years (because (302/301)^3<1.01), and in 31 years for a more realistic speed of 0.1c. From that point on, its only a matter of a few more centuries at most before we've used up whatever buffer we managed to make while we still expanded faster than our needs before the rot of stagnation catches up with us. That is, unless we manage to achieve stagnation without the rot.