[Kanegasi]

The gsub function accepts three arguments: the string, what to look for, and what to substitute.

msg=gsub(msg,find,replace)

or

msg=msg:gsub(find,replace)

This means that replace being an empty string, "", you are deleting what the function finds.

ChatFrame_AddMessageEventFilter("CHAT_MSG_BG_SYSTEM_ALLIANCE",function(_,_,msg,...)
    msg=msg:gsub(" If left unchallenged, the Alliance will control it in 1 minute!","")
    msg=msg:gsub(" control it in 1 minute!","")
    return nil,msg,...
end)

The above code only filters the blue text. To alter the red text, you need to filter the Horde event.

ChatFrame_AddMessageEventFilter("CHAT_MSG_BG_SYSTEM_HORDE",function(_,_,msg,...)
    msg=msg:gsub("The Horde Flag was picked up by ","")
    msg=msg:gsub("The Horde Flag was captured by ","")
    return nil,msg,...
end)

If I got the colors wrong, I apologize. The messages above may all be blue, it's been a long time since I played battlegrounds.

Edit:

Now that I think about it, you don't have to worry about the different events if you just filter all of them at once.

local filterfunc=function(_,_,msg,...)
    msg=msg:gsub(" If left unchallenged, the Alliance will control it in 1 minute!","")
    msg=msg:gsub(" control it in 1 minute!","")
    msg=msg:gsub("The Horde Flag was picked up by ","")
    msg=msg:gsub("The Horde Flag was captured by ","")
    return nil,msg,...
end
 
ChatFrame_AddMessageEventFilter("CHAT_MSG_BG_SYSTEM_ALLIANCE",filterfunc)
ChatFrame_AddMessageEventFilter("CHAT_MSG_BG_SYSTEM_HORDE",filterfunc)
ChatFrame_AddMessageEventFilter("CHAT_MSG_BG_SYSTEM_NEUTRAL",filterfunc)

This takes one filter function and applies it to all three BG announcement events.