Well first up - just to be pedantic but accurate - it isn't an error. I am also not totally sure how I can phrase it any better than it is already phrased.

Many WordPress plugins will want to create items on your server and the usual and standard place is under the 'wp-content' folder. As the message clearly states - the install tried and failed because the file permissions set on wp-content do not allow for it. This is most unusual but seems to be the case on your installation. Correcting the permissions - which most usually would be 755 on a folder and 644 on a file - will enable the install to be carried out.

It might also be the GUI/UID not being set right. These are ownership 'roles' that go hand in hand with the actual permissions. Rarely but every now and then we do seem to encounter a host that does not set the GUI/UID correctly - or perhaps I should say - not how WordPress would expect them to be. This will stop any attempt to create folders or files dead in the tracks.

To be honest - when it comes to this I always think the safest route is to ask your hosting support why folders can not be created?; if it is the gUID/UID? and can the wp-content folder be set up so any folders it spawns can not only be created but will carry to right set of permissions themselves? The chances are they will not let you change the UID/GUIDs in any case...

Weird. We simply use the most common test for this which is the php function is_writable() - and to copy from the manual...

Returns TRUE if the filename exists and is writable. The filename argument may be a directory name allowing you to check if a directory is writable.

Keep in mind that PHP may be accessing the file as the user id that the web server runs as (often 'nobody'). Safe mode limitations are not taken into account.

It must be objecting to something it is finding. Or perhaps 'Safe Mode' has this effect. I have never used it so can not comment from personal experience,