How do I unclog outside water drainage pipe?
I have a grate exactly like that one in my basement floor drain. You should be able to flip it up with a flat screwdriver or similar tool and pull it out of the way so you can run a snake down the drain.

I would start by using a shopvac to clear all the water above the drain. Then pull off the grate and start snaking. Impossible to guess how far you will need to go.

The big question is "what" is clogging in the drain. Since this is outside, it could be connected to your main drain and have "normal" drain clogs - e.g., lint, grease, etc. that a snake can relatively easily get through. But it is also possible that it is roots (which a snake can handle, but is sometimes quite a bit of rock) or even accumulated dirt/sand/gravel which may be really tough to break through. Only way to know is to try.