It depends on the country you're in, but in most (European) countries they will not allow you to enter the home stands while you are wearing a shirt/scarf/hat (well, really anything affiliated with the opposing team.
To go on from your given example, Manchester United, we get into English territory. If you are a bit into football you'll know that the English supporters' scene had a bit of rough time in history. But discussing that would lead us too far. To keep things short, away and home fans are kept as much apart as necessary and as much as they can. For English football that means completely separate stands.
Is this really the case?
As far as my limited experience in English grounds goes, their security measures are rather tight and maybe a bit too much anno 2015. But from the looks of it you're going rather as a neutral fan or a Manchester United fan to such game.
If you're buying tickets through a website like that one, tickets will be in the home sections of the stadium and you will have no problem entering the stadium with some Manchester United fan gear. So it has a double answer: "Yes" as an away supporter, "No" as a home supporter.
What are they trying to accomplish?
Well, just to keep fans separated. What the site is trying to accomplish is just not be in any danger to complaints from the club selling tickets to other teams' supporters. They are just covering their asses with a little message.
Don't they already separate the supporters within the stadium with security and/or walls?
Yes, but no system is without error.
An exception on not allowing obvious away supporters in home stand is Germany, apart from the stand with the hardcore fans you will most likely be able to enter with for instance wearing a shirt from the away team in the home stand.
For major tournaments (World Cup, European Championship, Champions League Final etc) only a small part is sold directly to the two teams supporters group directly, the rest is sold in a general sale as neutral tickets, this is were mixed supporter situations are created. Most tickets will be bought up by local fans and off course supporters of the two playing teams.