In the method that you're using you need to make two measurements because you are using the relation $\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}$ so you need to measure the object and image distances, $d_o$ and $d_i$. Another common technique for measuring the focal length is to find the image distance for an object that is distant enough that the object distance can be considered infinite. In that case $f=d_i$.
To answer why you can see a real image it helps to remember that images look like objects because light comes from an images in the same way as it comes from an object. The real image is formed in front of the lens, that is to say on the same side of the lens as you and the opposite side compared to object A. Rays of light from the object pass through the lens, converge at the point of the image and then diverge again. 586
You speak of seeing the image "in the lens" and it's true the image has to be lined between your eye and the lens but if you look carefully and move you head slightly while looking at the image you will find that it behaves as if it were in front of the lens, rather behind it as in a virtual image.
You could hold a screen (piece of paper) at the point of where the image is formed and you might be able to see the real image projected on it though probably not. This would work more reliably if you were using a candle or some other luminous object. With respect to the retina of you eye, what is happening there is that your eye's lens (with the cornea) is forming a real image on your retina. This process is the same regardless of whether you are looking at a real image, or a virtual image or indeed the original object itself.
In the lab procedure you are trying to place object B so that it coincides with the image of object A. The idea is if get it in the right spot you can move your head around a bit and the object A's image and object B should appear to remain together. If they seem to move relative to each other as you change your perspective then you know that they are not yet in the same place.
This may seem like a complicated way to measure the focal length but it does give you some experience of working with real images. 530