If grass is too long when you take a reel mower to it, it just folds over. This is made even worse when it is thick. After making sure the blades are sharp, you will probably just need to take multiple passes at it.
We own two reel mowers and the way we know we need to sharpen one is when the same grass responds like that on one mower and normally on another. But I wouldn't let the grass get that high. (Yes, I know that's a frustrating challenge when it keeps raining and the grass keeps growing.) The key to success with reel mowing is to do it often. I find that easy enough to do because the mower is light weight and easy to use, so I can do one bit of lawn that's getting long without full on Mowing The Lawn as some kind of production - I just mow the part that's getting long. (We have an acre and a half, but only about half it is lawn.)
If it won't stop raining and the grass won't stop growing and when you get to it, it's like that, well you just keep going over it. Go in opposing directions, that may help a little. When it stands back up, go take another pass at it. Don't worry about what's a weed and what's grass. It all needs to get cut. For more tips, see https://gardening.stackexchange.com/a/5014/754
Update: I was super frustrated with my reel mower yesterday because it was doing that same thing you asked about. I was going over and over the same ground and it would produce clippings, but the lawn still looked unmowed. I sharpened it (which took about an hour) but the result was the same. But while interacting with the reel to sharpen it I came to realize that the "cutting bar" - the straight thing the reel blades need to contact with to slice the grass like scissors - needed to be raised at one end. The blades were not contacting it on the entire left side of the mower. The grass was not getting cut because the blades were not touching the bar. I adjusted the bar and presto! When I mow a piece of lawn, it stays mowed. Adjusting the bar is much quicker than sharpening so check for that first.
To check, put the mower on its back and spin the reel (carefully) with your hand. If it moves really easily and then a bit stiff, really easy and then stiff, look closely at the bar. Are the blades touching along all the length of the bar as they turn? If not, find your instructions and adjust it.