At the leading edge of that black plastic cover, which incorporates fresh air vents for your car's interior, there are two drain pipes which run down, against the firewall. These drains commonly get clogged.

There is an additional drain, however, underneath that black plastic cover, which can also get clogged and exhibit symptoms exactly as you describe. You need to remove that black plastic cover and clean out that hidden drain.

This exact same thing happened to my car. When the problem gets bad, water will start spilling out on the passenger-side floor, and then you've got a real mess.