Density altitude ("DA" in pilot lingo) affects many things, but the factor that is usually most critical is an aircraft's ability to get off a runway and climb. If it's very hot out, that can require a much longer takeoff roll to get off the ground, and dramatically reduced climb performance.

If the runway in use is short, or if there's high terrain (mountains) close-in around an airport, density altitude can be extremely dangerous due to degraded aircraft performance.

DA is actually one of the leading causes of fatal aircraft accidents in the US west, but those accidents almost always involve smaller airplanes that are more marginally powered. Commercial jets typically have plenty of performance to overcome DA, but if conditions are right (higher altitude, high temperatures, shorter runways, full loads) it can make enough difference to be dangerous.